:: Predicate Calculus for Boolean Valued Functions, { VI }
:: by Shunichi Kobayashi
::
:: Received October 19, 1999
:: Copyright (c) 1999-2026 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies XBOOLE_0, SUBSET_1, PARTIT1, EQREL_1, TARSKI, SETFAM_1, FUNCOP_1,
RELAT_1, ZFMISC_1, FUNCT_1, FUNCT_4, BVFUNC_2;
notations TARSKI, XBOOLE_0, ENUMSET1, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1,
SETFAM_1, EQREL_1, FUNCOP_1, PARTIT1, BVFUNC_1, BVFUNC_2, FUNCT_4;
constructors ENUMSET1, SETFAM_1, FUNCT_4, BVFUNC_1, BVFUNC_2, FUNCOP_1;
registrations XBOOLE_0, SUBSET_1, FUNCOP_1, FUNCT_4, EQREL_1;
requirements SUBSET, BOOLE;
definitions TARSKI;
equalities FUNCOP_1, FUNCT_4;
expansions TARSKI;
theorems TARSKI, FUNCT_1, SETFAM_1, EQREL_1, ZFMISC_1, PARTIT1, BVFUNC_2,
BVFUNC11, ENUMSET1, FUNCOP_1, FUNCT_4, XBOOLE_0, XBOOLE_1, FUNCT_7;
begin :: Chap. 1 Preliminaries
reserve Y for non empty set,
G for Subset of PARTITIONS(Y),
A,B,C,D,E,F for a_partition of Y;
theorem Th1:
for z being Element of Y, PA,PB being a_partition of Y holds
EqClass(z,PA '/\' PB) = EqClass(z,PA) /\ EqClass(z,PB)
proof
let z be Element of Y, PA,PB be a_partition of Y;
A1: EqClass(z,PA) /\ EqClass(z,PB) c= EqClass(z,PA '/\' PB)
proof
set Z=EqClass(z,PA '/\' PB);
let x be object;
assume
A2: x in EqClass(z,PA) /\ EqClass(z,PB);
then reconsider x as Element of Y;
A3: x in EqClass(x,PA) by EQREL_1:def 6;
x in EqClass(z,PA) by A2,XBOOLE_0:def 4;
then
A4: EqClass(x,PA) meets EqClass(z,PA) by A3,XBOOLE_0:3;
A5: x in EqClass(x,PB) by EQREL_1:def 6;
PA '/\' PB = INTERSECTION(PA,PB) \ {{}} by PARTIT1:def 4;
then Z in INTERSECTION(PA,PB) by XBOOLE_0:def 5;
then consider X,Y being set such that
A6: X in PA and
A7: Y in PB and
A8: Z = X /\ Y by SETFAM_1:def 5;
A9: z in X /\ Y by A8,EQREL_1:def 6;
then z in EqClass(z,PB) & z in Y by EQREL_1:def 6,XBOOLE_0:def 4;
then Y meets EqClass(z,PB) by XBOOLE_0:3;
then
A10: Y=EqClass(z,PB) by A7,EQREL_1:def 4;
x in EqClass(z,PB) by A2,XBOOLE_0:def 4;
then
A11: EqClass(x,PB) meets EqClass(z,PB) by A5,XBOOLE_0:3;
z in EqClass(z,PA) & z in X by A9,EQREL_1:def 6,XBOOLE_0:def 4;
then X meets EqClass(z,PA) by XBOOLE_0:3;
then X=EqClass(z,PA) by A6,EQREL_1:def 4;
then
A12: X=EqClass(x,PA) by A4,EQREL_1:41;
x in EqClass(x,PA) /\ EqClass(x,PB) by A3,A5,XBOOLE_0:def 4;
hence thesis by A11,A8,A10,A12,EQREL_1:41;
end;
EqClass(z,PA '/\' PB) c= EqClass(z,PA) /\ EqClass(z,PB)
proof
let x be object;
A13: EqClass(z,PA '/\' PB) c= EqClass(z,PA) & EqClass(z,PA '/\' PB) c=
EqClass(z, PB) by BVFUNC11:3;
assume x in EqClass(z,PA '/\' PB);
hence thesis by A13,XBOOLE_0:def 4;
end;
hence thesis by A1,XBOOLE_0:def 10;
end;
theorem
G={A,B} & A<>B implies '/\' G = A '/\' B
proof
assume that
A1: G={A,B} and
A2: A<>B;
A3: A '/\' B c= '/\' G
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume
A4: x in A '/\' B;
then
A5: x<>{} by EQREL_1:def 4;
x in INTERSECTION(A,B) \ {{}} by A4,PARTIT1:def 4;
then consider a,b being set such that
A6: a in A and
A7: b in B and
A8: x = a /\ b by SETFAM_1:def 5;
set h0=(A,B) --> (a,b);
A9: rng((A,B) --> (a,b)) = {a,b} by A2,FUNCT_4:64;
rng h0 c= bool Y
proof
let y be object;
assume
A10: y in rng h0;
now
per cases by A9,A10,TARSKI:def 2;
case
y=a;
hence thesis by A6;
end;
case
y=b;
hence thesis by A7;
end;
end;
hence thesis;
end;
then reconsider F=rng h0 as Subset-Family of Y;
A11: xx c= Intersect F
proof
let u be object;
assume
A12: u in xx;
for y be set holds y in F implies u in y
proof
let y be set;
assume
A13: y in F;
now
per cases by A9,A13,TARSKI:def 2;
case
y=a;
hence thesis by A8,A12,XBOOLE_0:def 4;
end;
case
y=b;
hence thesis by A8,A12,XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u in meet F by A9,SETFAM_1:def 1;
hence thesis by A9,SETFAM_1:def 9;
end;
A14: for d being set st d in G holds h0.d in d
proof
let d be set;
assume
A15: d in G;
now
per cases by A1,A15,TARSKI:def 2;
case
d=A;
hence thesis by A2,A6,FUNCT_4:63;
end;
case
d=B;
hence thesis by A7,FUNCT_4:63;
end;
end;
hence thesis;
end;
A16: rng h0 = {a,b} by A2,FUNCT_4:64;
Intersect F c= xx
proof
let u be object;
assume
A17: u in Intersect F;
A18: a in {a,b} by TARSKI:def 2;
then a in F by A2,FUNCT_4:64;
then
A19: Intersect F = meet F by SETFAM_1:def 9;
b in {a,b} by TARSKI:def 2;
then
A20: u in b by A16,A17,A19,SETFAM_1:def 1;
u in a by A16,A17,A18,A19,SETFAM_1:def 1;
hence thesis by A8,A20,XBOOLE_0:def 4;
end;
then dom((A,B) --> (a,b)) = {A,B} & x = Intersect F by A11,FUNCT_4:62
,XBOOLE_0:def 10;
hence thesis by A1,A14,A5,BVFUNC_2:def 1;
end;
'/\' G c= A '/\' B
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume x in '/\' G;
then consider h being Function, F being Subset-Family of Y such that
A21: dom h=G and
A22: rng h = F and
A23: for d being set st d in G holds h.d in d and
A24: x=Intersect F and
A25: x<>{} by BVFUNC_2:def 1;
A26: not x in {{}} by A25,TARSKI:def 1;
A in dom h by A1,A21,TARSKI:def 2;
then
A27: h.A in rng h by FUNCT_1:def 3;
A28: h.A /\ h.B c= xx
proof
let m be object;
assume
A29: m in h.A /\ h.B;
A30: rng h c= {h.A,h.B}
proof
let u be object;
assume u in rng h;
then consider x1 being object such that
A31: x1 in dom h and
A32: u = h.x1 by FUNCT_1:def 3;
now
per cases by A1,A21,A31,TARSKI:def 2;
case
x1=A;
hence thesis by A32,TARSKI:def 2;
end;
case
x1=B;
hence thesis by A32,TARSKI:def 2;
end;
end;
hence thesis;
end;
for y being set holds y in rng h implies m in y
proof
let y be set;
assume
A33: y in rng h;
now
per cases by A30,A33,TARSKI:def 2;
case
y=h.A;
hence thesis by A29,XBOOLE_0:def 4;
end;
case
y=h.B;
hence thesis by A29,XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then m in meet (rng h) by A27,SETFAM_1:def 1;
hence thesis by A22,A24,A27,SETFAM_1:def 9;
end;
B in G by A1,TARSKI:def 2;
then
A34: h.B in B by A23;
A in G by A1,TARSKI:def 2;
then
A35: h.A in A by A23;
B in dom h by A1,A21,TARSKI:def 2;
then
A36: h.B in rng h by FUNCT_1:def 3;
xx c= h.A /\ h.B
proof
let m be object;
assume m in xx;
then m in meet (rng h) by A22,A24,A27,SETFAM_1:def 9;
then m in h.A & m in h.B by A27,A36,SETFAM_1:def 1;
hence thesis by XBOOLE_0:def 4;
end;
then h.A /\ h.B = x by A28,XBOOLE_0:def 10;
then x in INTERSECTION(A,B) by A35,A34,SETFAM_1:def 5;
then x in INTERSECTION(A,B) \ {{}} by A26,XBOOLE_0:def 5;
hence thesis by PARTIT1:def 4;
end;
hence thesis by A3,XBOOLE_0:def 10;
end;
Lm1: for f being Function, C,D,c,d being object st C<>D
holds (f +* (C .--> c) +*(D .--> d)).C = c
proof
let f be Function;
let C,D,c,d be object;
set h = f +* (C .--> c) +* (D .--> d);
assume C<>D;
then not C in dom (D .--> d) by TARSKI:def 1;
then
A2: h.C=(f +* (C .--> c)).C by FUNCT_4:11;
C in dom (C .--> c) by TARSKI:def 1;
hence h.C=(C .--> c).C by A2,FUNCT_4:13
.= c by FUNCOP_1:72;
end;
Lm2: for B,C,D,b,c,d being object, h being Function
st h = (B,C,D) --> (b,c,d)
holds rng h = {h.B,h.C,h.D}
proof
let B,C,D,b,c,d be object, h be Function;
assume h = (B,C,D) --> (b,c,d);
then
A1: dom h = {B,C,D} by FUNCT_4:128;
then
A2: B in dom h by ENUMSET1:def 1;
A3: rng h c= {h.B,h.C,h.D}
proof
let t be object;
assume t in rng h;
then consider x1 being object such that
A4: x1 in dom h and
A5: t = h.x1 by FUNCT_1:def 3;
now
per cases by A1,A4,ENUMSET1:def 1;
case
x1=D;
hence thesis by A5,ENUMSET1:def 1;
end;
case
x1=B;
hence thesis by A5,ENUMSET1:def 1;
end;
case
x1=C;
hence thesis by A5,ENUMSET1:def 1;
end;
end;
hence thesis;
end;
A6: C in dom h by A1,ENUMSET1:def 1;
A7: D in dom h by A1,ENUMSET1:def 1;
{h.B,h.C,h.D} c= rng h
proof
let t be object;
assume
A8: t in {h.B,h.C,h.D};
now
per cases by A8,ENUMSET1:def 1;
case
t=h.D;
hence thesis by A7,FUNCT_1:def 3;
end;
case
t=h.B;
hence thesis by A2,FUNCT_1:def 3;
end;
case
t=h.C;
hence thesis by A6,FUNCT_1:def 3;
end;
end;
hence thesis;
end;
hence thesis by A3,XBOOLE_0:def 10;
end;
theorem
G={B,C,D} & B<>C & C<>D & D<>B implies '/\' G = B '/\' C '/\' D
proof
assume that
A1: G={B,C,D} and
A2: B<>C and
A3: C<>D and
A4: D<>B;
A5: B '/\' C '/\' D c= '/\' G
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume
A6: x in B '/\' C '/\' D;
then
A7: x<>{} by EQREL_1:def 4;
x in INTERSECTION(B '/\' C,D) \ {{}} by A6,PARTIT1:def 4;
then consider a,d being set such that
A8: a in B '/\' C and
A9: d in D and
A10: x = a /\ d by SETFAM_1:def 5;
a in INTERSECTION(B,C) \ {{}} by A8,PARTIT1:def 4;
then consider b,c being set such that
A11: b in B and
A12: c in C and
A13: a = b /\ c by SETFAM_1:def 5;
set h = (B,C,D) --> (b,c,d);
A14: rng h = {h.B,h.C,h.D} by Lm2
.= {h.D,h.B,h.C} by ENUMSET1:59;
A15: h.D = d by FUNCT_7:94;
rng h c= bool Y
proof
let t be object;
assume
A16: t in rng h;
now
per cases by A14,A16,ENUMSET1:def 1;
case
t=h.D;
hence thesis by A9,A15;
end;
case
t=h.B;
then t=b by A2,A4,FUNCT_4:134;
hence thesis by A11;
end;
case
t=h.C;
then t=c by A3,Lm1;
hence thesis by A12;
end;
end;
hence thesis;
end;
then reconsider F=rng h as Subset-Family of Y;
A17: h.C = c by A3,Lm1;
A18: for p being set st p in G holds h.p in p
proof
let p be set;
assume
A19: p in G;
now
per cases by A1,A19,ENUMSET1:def 1;
case
p=D;
hence thesis by A9,FUNCT_7:94;
end;
case
p=B;
hence thesis by A2,A4,A11,FUNCT_4:134;
end;
case
p=C;
hence thesis by A3,A12,Lm1;
end;
end;
hence thesis;
end;
A20: h.B = b by A2,A4,FUNCT_4:134;
A21: xx c= Intersect F
proof
let u be object;
assume
A22: u in xx;
for y be set holds y in F implies u in y
proof
let y be set;
assume
A23: y in F;
now
per cases by A14,A23,ENUMSET1:def 1;
case
y=h.D;
hence thesis by A10,A15,A22,XBOOLE_0:def 4;
end;
case
A24: y=h.B;
u in b /\ (c /\ d) by A10,A13,A22,XBOOLE_1:16;
hence thesis by A20,A24,XBOOLE_0:def 4;
end;
case
A25: y=h.C;
u in c /\ (b /\ d) by A10,A13,A22,XBOOLE_1:16;
hence thesis by A17,A25,XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u in meet F by A14,SETFAM_1:def 1;
hence thesis by A14,SETFAM_1:def 9;
end;
A26: dom h = {B,C,D} by FUNCT_4:128;
then D in dom h by ENUMSET1:def 1;
then
A27: rng h <> {} by FUNCT_1:3;
Intersect F c= xx
proof
let t be object;
assume t in Intersect F;
then
A28: t in meet (rng h) by A27,SETFAM_1:def 9;
h.C in {h.D,h.B,h.C} by ENUMSET1:def 1;
then t in h.C by A14,A28,SETFAM_1:def 1;
then
A29: t in c by A3,Lm1;
h.B in {h.D,h.B,h.C} by ENUMSET1:def 1;
then t in h.B by A14,A28,SETFAM_1:def 1;
then t in b by A2,A4,FUNCT_4:134;
then
A30: t in b /\ c by A29,XBOOLE_0:def 4;
h.D in {h.D,h.B,h.C} by ENUMSET1:def 1;
then t in h.D by A14,A28,SETFAM_1:def 1;
hence thesis by A10,A13,A15,A30,XBOOLE_0:def 4;
end;
then x = Intersect F by A21,XBOOLE_0:def 10;
hence thesis by A1,A26,A18,A7,BVFUNC_2:def 1;
end;
'/\' G c= B '/\' C '/\' D
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume x in '/\' G;
then consider h being Function, F being Subset-Family of Y such that
A31: dom h=G and
A32: rng h = F and
A33: for d being set st d in G holds h.d in d and
A34: x=Intersect F and
A35: x<>{} by BVFUNC_2:def 1;
D in dom h by A1,A31,ENUMSET1:def 1;
then
A36: h.D in rng h by FUNCT_1:def 3;
set m=h.B /\ h.C;
B in dom h by A1,A31,ENUMSET1:def 1;
then
A37: h.B in rng h by FUNCT_1:def 3;
C in dom h by A1,A31,ENUMSET1:def 1;
then
A38: h.C in rng h by FUNCT_1:def 3;
A39: xx c= h.B /\ h.C /\ h.D
proof
let m be object;
assume m in xx;
then
A40: m in meet (rng h) by A32,A34,A37,SETFAM_1:def 9;
then m in h.B & m in h.C by A37,A38,SETFAM_1:def 1;
then
A41: m in h.B /\ h.C by XBOOLE_0:def 4;
m in h.D by A36,A40,SETFAM_1:def 1;
hence thesis by A41,XBOOLE_0:def 4;
end;
then m<>{} by A35;
then
A42: not m in {{}} by TARSKI:def 1;
D in G by A1,ENUMSET1:def 1;
then
A43: h.D in D by A33;
A44: not x in {{}} by A35,TARSKI:def 1;
C in G by A1,ENUMSET1:def 1;
then
A45: h.C in C by A33;
B in G by A1,ENUMSET1:def 1;
then h.B in B by A33;
then m in INTERSECTION(B,C) by A45,SETFAM_1:def 5;
then m in INTERSECTION(B,C) \ {{}} by A42,XBOOLE_0:def 5;
then
A46: m in B '/\' C by PARTIT1:def 4;
h.B /\ h.C /\ h.D c= xx
proof
let m be object;
assume
A47: m in h.B /\ h.C /\ h.D;
then
A48: m in h.B /\ h.C by XBOOLE_0:def 4;
A49: rng h c= {h.B,h.C,h.D}
proof
let u be object;
assume u in rng h;
then consider x1 being object such that
A50: x1 in dom h and
A51: u = h.x1 by FUNCT_1:def 3;
now
per cases by A1,A31,A50,ENUMSET1:def 1;
case
x1=B;
hence thesis by A51,ENUMSET1:def 1;
end;
case
x1=C;
hence thesis by A51,ENUMSET1:def 1;
end;
case
x1=D;
hence thesis by A51,ENUMSET1:def 1;
end;
end;
hence thesis;
end;
for y being set holds y in rng h implies m in y
proof
let y be set;
assume
A52: y in rng h;
now
per cases by A49,A52,ENUMSET1:def 1;
case
y=h.B;
hence thesis by A48,XBOOLE_0:def 4;
end;
case
y=h.C;
hence thesis by A48,XBOOLE_0:def 4;
end;
case
y=h.D;
hence thesis by A47,XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then m in meet (rng h) by A37,SETFAM_1:def 1;
hence thesis by A32,A34,A37,SETFAM_1:def 9;
end;
then (h.B /\ h.C) /\ h.D = x by A39,XBOOLE_0:def 10;
then x in INTERSECTION(B '/\' C,D) by A43,A46,SETFAM_1:def 5;
then x in INTERSECTION(B '/\' C,D) \ {{}} by A44,XBOOLE_0:def 5;
hence thesis by PARTIT1:def 4;
end;
hence thesis by A5,XBOOLE_0:def 10;
end;
theorem Th4:
G={A,B,C} & A<>B & C<>A implies CompF(A,G) = B '/\' C
proof
assume that
A1: G={A,B,C} and
A2: A<>B and
A3: C<>A;
per cases;
suppose
A4: B = C;
G = {B,C,A} by A1,ENUMSET1:59
.= {B,A} by A4,ENUMSET1:30;
hence CompF(A,G) = B by A2,BVFUNC11:7
.= B '/\' C by A4,PARTIT1:13;
end;
suppose
A5: B <> C;
A6: G \ {A}={A} \/ {B,C} \ {A} by A1,ENUMSET1:2
.= ({A} \ {A}) \/ ({B,C} \ {A}) by XBOOLE_1:42;
( not B in {A})& not C in {A} by A2,A3,TARSKI:def 1;
then
A7: G \ {A} = ({A} \ {A}) \/ {B,C} by A6,ZFMISC_1:63
.= {} \/ {B,C} by XBOOLE_1:37
.= {B,C};
A8: '/\' (G \ {A}) c= B '/\' C
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume x in '/\' (G \ {A});
then consider h being Function, F being Subset-Family of Y such that
A9: dom h=(G \ {A}) and
A10: rng h = F and
A11: for d being set st d in (G \ {A}) holds h.d in d and
A12: x=Intersect F and
A13: x<>{} by BVFUNC_2:def 1;
A14: not x in {{}} by A13,TARSKI:def 1;
B in dom h by A7,A9,TARSKI:def 2;
then
A15: h.B in rng h by FUNCT_1:def 3;
A16: h.B /\ h.C c= xx
proof
let m be object;
assume
A17: m in h.B /\ h.C;
A18: rng h c= {h.B,h.C}
proof
let u be object;
assume u in rng h;
then consider x1 being object such that
A19: x1 in dom h and
A20: u = h.x1 by FUNCT_1:def 3;
now
per cases by A7,A9,A19,TARSKI:def 2;
case
x1=B;
hence thesis by A20,TARSKI:def 2;
end;
case
x1=C;
hence thesis by A20,TARSKI:def 2;
end;
end;
hence thesis;
end;
for y being set holds y in rng h implies m in y
proof
let y be set;
assume
A21: y in rng h;
now
per cases by A18,A21,TARSKI:def 2;
case
y=h.B;
hence thesis by A17,XBOOLE_0:def 4;
end;
case
y=h.C;
hence thesis by A17,XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then m in meet (rng h) by A15,SETFAM_1:def 1;
hence thesis by A10,A12,A15,SETFAM_1:def 9;
end;
C in (G \ {A}) by A7,TARSKI:def 2;
then
A22: h.C in C by A11;
B in (G \ {A}) by A7,TARSKI:def 2;
then
A23: h.B in B by A11;
C in dom h by A7,A9,TARSKI:def 2;
then
A24: h.C in rng h by FUNCT_1:def 3;
xx c= h.B /\ h.C
proof
let m be object;
assume m in xx;
then m in meet (rng h) by A10,A12,A15,SETFAM_1:def 9;
then m in h.B & m in h.C by A15,A24,SETFAM_1:def 1;
hence thesis by XBOOLE_0:def 4;
end;
then h.B /\ h.C = x by A16,XBOOLE_0:def 10;
then x in INTERSECTION(B,C) by A23,A22,SETFAM_1:def 5;
then x in INTERSECTION(B,C) \ {{}} by A14,XBOOLE_0:def 5;
hence thesis by PARTIT1:def 4;
end;
A25: B '/\' C c= '/\' (G \ {A})
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume
A26: x in B '/\' C;
then
A27: x<>{} by EQREL_1:def 4;
x in INTERSECTION(B,C) \ {{}} by A26,PARTIT1:def 4;
then consider a,b being set such that
A28: a in B and
A29: b in C and
A30: x = a /\ b by SETFAM_1:def 5;
set h0=(B,C) --> (a,b);
A31: dom h0 = (G \ {A}) by A7,FUNCT_4:62;
A32: rng h0 = {a,b} by A5,FUNCT_4:64;
rng h0 c= bool Y
proof
let y be object;
assume
A33: y in rng h0;
now
per cases by A32,A33,TARSKI:def 2;
case
y=a;
hence thesis by A28;
end;
case
y=b;
hence thesis by A29;
end;
end;
hence thesis;
end;
then reconsider F=rng h0 as Subset-Family of Y;
A34: xx c= Intersect F
proof
let u be object;
assume
A35: u in xx;
for y be set holds y in F implies u in y
proof
let y be set;
assume
A36: y in F;
now
per cases by A32,A36,TARSKI:def 2;
case
y=a;
hence thesis by A30,A35,XBOOLE_0:def 4;
end;
case
y=b;
hence thesis by A30,A35,XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u in meet F by A32,SETFAM_1:def 1;
hence thesis by A32,SETFAM_1:def 9;
end;
A37: for d being set st d in (G \ {A}) holds h0.d in d
proof
let d be set;
assume
A38: d in (G \ {A});
now
per cases by A7,A38,TARSKI:def 2;
case
d=B;
hence thesis by A5,A28,FUNCT_4:63;
end;
case
d=C;
hence thesis by A29,FUNCT_4:63;
end;
end;
hence thesis;
end;
Intersect F c= xx
proof
let u be object;
assume
A39: u in Intersect F;
A40: Intersect F = meet F by A32,SETFAM_1:def 9;
b in F by A32,TARSKI:def 2;
then
A41: u in b by A39,A40,SETFAM_1:def 1;
a in F by A32,TARSKI:def 2;
then u in a by A39,A40,SETFAM_1:def 1;
hence thesis by A30,A41,XBOOLE_0:def 4;
end;
then x = Intersect F by A34,XBOOLE_0:def 10;
hence thesis by A31,A37,A27,BVFUNC_2:def 1;
end;
CompF(A,G)='/\' (G \ {A}) by BVFUNC_2:def 7;
hence thesis by A25,A8,XBOOLE_0:def 10;
end;
end;
theorem Th5:
G={A,B,C} & A<>B & B<>C implies CompF(B,G) = C '/\' A
proof
{A,B,C} = {B,C,A} by ENUMSET1:59;
hence thesis by Th4;
end;
theorem
G={A,B,C} & B<>C & C<>A implies CompF(C,G) = A '/\' B
proof
{A,B,C} = {C,A,B} by ENUMSET1:59;
hence thesis by Th4;
end;
theorem Th7:
G={A,B,C,D} & A<>B & A<>C & A<>D implies CompF(A,G) = B '/\' C '/\' D
proof
assume that
A1: G={A,B,C,D} and
A2: A<>B and
A3: A<>C and
A4: A<>D;
per cases;
suppose
A5: B = C;
then G = {B,B,A,D} by A1,ENUMSET1:71
.= {B,A,D} by ENUMSET1:31
.= {A,B,D} by ENUMSET1:58;
hence CompF(A,G)= B '/\' D by A2,A4,Th4
.= B '/\' C '/\' D by A5,PARTIT1:13;
end;
suppose
A6: B = D;
then G = {B,B,A,C} by A1,ENUMSET1:69
.= {B,A,C} by ENUMSET1:31
.= {A,B,C} by ENUMSET1:58;
hence CompF(A,G) = B '/\' C by A2,A3,Th4
.= B '/\' D '/\' C by A6,PARTIT1:13
.= B '/\' C '/\' D by PARTIT1:14;
end;
suppose
A7: C = D;
then G = {C,C,A,B} by A1,ENUMSET1:73
.= {C,A,B} by ENUMSET1:31
.= {A,B,C} by ENUMSET1:59;
hence CompF(A,G) = B '/\' C by A2,A3,Th4
.= B '/\' (C '/\' D) by A7,PARTIT1:13
.= B '/\' C '/\' D by PARTIT1:14;
end;
suppose
A8: B<>C & B<>D & C<>D;
G \ {A}={A} \/ {B,C,D} \ {A} by A1,ENUMSET1:4;
then
A9: G \ {A} = ({A} \ {A}) \/ ({B,C,D} \ {A}) by XBOOLE_1:42;
A10: not B in {A} by A2,TARSKI:def 1;
A11: ( not C in {A})& not D in {A} by A3,A4,TARSKI:def 1;
{B,C,D} \ {A} = ({B} \/ {C,D}) \ {A} by ENUMSET1:2
.= ({B} \ {A}) \/ ({C,D} \ {A}) by XBOOLE_1:42
.= ({B} \ {A}) \/ ({C,D}) by A11,ZFMISC_1:63
.= {B} \/ {C,D} by A10,ZFMISC_1:59
.= {B,C,D} by ENUMSET1:2;
then
A12: G \ {A} = {} \/ {B,C,D} by A9,XBOOLE_1:37
.= {B,C,D};
A13: B '/\' C '/\' D c= '/\' (G \ {A})
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume
A14: x in B '/\' C '/\' D;
then
A15: x<>{} by EQREL_1:def 4;
x in INTERSECTION(B '/\' C,D) \ {{}} by A14,PARTIT1:def 4;
then consider a,d being set such that
A16: a in B '/\' C and
A17: d in D and
A18: x = a /\ d by SETFAM_1:def 5;
a in INTERSECTION(B,C) \ {{}} by A16,PARTIT1:def 4;
then consider b,c being set such that
A19: b in B and
A20: c in C and
A21: a = b /\ c by SETFAM_1:def 5;
set h = (B,C,D) --> (b,c,d);
A22: h.D = d by FUNCT_7:94;
A23: h.C = c by A8,Lm1;
A24: rng h = {h.B,h.C,h.D} by Lm2
.= {h.D,h.B,h.C} by ENUMSET1:59;
A25: h.B = b by A8,FUNCT_4:134;
rng h c= bool Y
proof
let t be object;
assume
A26: t in rng h;
now
per cases by A24,A26,ENUMSET1:def 1;
case
t=h.D;
hence thesis by A17,A22;
end;
case
t=h.B;
hence thesis by A19,A25;
end;
case
t=h.C;
hence thesis by A20,A23;
end;
end;
hence thesis;
end;
then reconsider F=rng h as Subset-Family of Y;
A27: xx c= Intersect F
proof
let u be object;
assume
A28: u in xx;
for y be set holds y in F implies u in y
proof
let y be set;
assume
A29: y in F;
now
per cases by A24,A29,ENUMSET1:def 1;
case
y=h.D;
hence thesis by A18,A22,A28,XBOOLE_0:def 4;
end;
case
A30: y=h.B;
u in b /\ (c /\ d) by A18,A21,A28,XBOOLE_1:16;
hence thesis by A25,A30,XBOOLE_0:def 4;
end;
case
A31: y=h.C;
u in c /\ (b /\ d) by A18,A21,A28,XBOOLE_1:16;
hence thesis by A23,A31,XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u in meet F by A24,SETFAM_1:def 1;
hence thesis by A24,SETFAM_1:def 9;
end;
A32: for p being set st p in (G \ {A}) holds h.p in p
proof
let p be set;
assume
A33: p in (G \ {A});
now
per cases by A12,A33,ENUMSET1:def 1;
case
p=D;
hence thesis by A17,FUNCT_7:94;
end;
case
p=B;
hence thesis by A8,A19,FUNCT_4:134;
end;
case
p=C;
hence thesis by A8,A20,Lm1;
end;
end;
hence thesis;
end;
A34: dom h = {B,C,D} by FUNCT_4:128;
then D in dom h by ENUMSET1:def 1;
then
A35: rng h <> {} by FUNCT_1:3;
Intersect F c= xx
proof
let t be object;
assume t in Intersect F;
then
A36: t in meet (rng h) by A35,SETFAM_1:def 9;
h.D in rng h by A24,ENUMSET1:def 1;
then
A37: t in h.D by A36,SETFAM_1:def 1;
h.C in rng h by A24,ENUMSET1:def 1;
then
A38: t in h.C by A36,SETFAM_1:def 1;
h.B in rng h by A24,ENUMSET1:def 1;
then t in h.B by A36,SETFAM_1:def 1;
then t in b /\ c by A25,A23,A38,XBOOLE_0:def 4;
hence thesis by A18,A21,A22,A37,XBOOLE_0:def 4;
end;
then x = Intersect F by A27,XBOOLE_0:def 10;
hence thesis by A12,A34,A32,A15,BVFUNC_2:def 1;
end;
'/\' (G \ {A}) c= B '/\' C '/\' D
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume x in '/\' (G \ {A});
then consider h being Function, F being Subset-Family of Y such that
A39: dom h=(G \ {A}) and
A40: rng h = F and
A41: for d being set st d in (G \ {A}) holds h.d in d and
A42: x=Intersect F and
A43: x<>{} by BVFUNC_2:def 1;
D in dom h by A12,A39,ENUMSET1:def 1;
then
A44: h.D in rng h by FUNCT_1:def 3;
set m=h.B /\ h.C;
B in dom h by A12,A39,ENUMSET1:def 1;
then
A45: h.B in rng h by FUNCT_1:def 3;
C in dom h by A12,A39,ENUMSET1:def 1;
then
A46: h.C in rng h by FUNCT_1:def 3;
A47: xx c= h.B /\ h.C /\ h.D
proof
let m be object;
assume m in xx;
then
A48: m in meet (rng h) by A40,A42,A45,SETFAM_1:def 9;
then m in h.B & m in h.C by A45,A46,SETFAM_1:def 1;
then
A49: m in h.B /\ h.C by XBOOLE_0:def 4;
m in h.D by A44,A48,SETFAM_1:def 1;
hence thesis by A49,XBOOLE_0:def 4;
end;
then m<>{} by A43;
then
A50: not m in {{}} by TARSKI:def 1;
D in (G \ {A}) by A12,ENUMSET1:def 1;
then
A51: h.D in D by A41;
A52: not x in {{}} by A43,TARSKI:def 1;
C in (G \ {A}) by A12,ENUMSET1:def 1;
then
A53: h.C in C by A41;
B in (G \ {A}) by A12,ENUMSET1:def 1;
then h.B in B by A41;
then m in INTERSECTION(B,C) by A53,SETFAM_1:def 5;
then m in INTERSECTION(B,C) \ {{}} by A50,XBOOLE_0:def 5;
then
A54: m in B '/\' C by PARTIT1:def 4;
h.B /\ h.C /\ h.D c= xx
proof
let m be object;
assume
A55: m in h.B /\ h.C /\ h.D;
then
A56: m in h.B /\ h.C by XBOOLE_0:def 4;
A57: rng h c= {h.B,h.C,h.D}
proof
let u be object;
assume u in rng h;
then consider x1 being object such that
A58: x1 in dom h and
A59: u = h.x1 by FUNCT_1:def 3;
now
per cases by A12,A39,A58,ENUMSET1:def 1;
case
x1=B;
hence thesis by A59,ENUMSET1:def 1;
end;
case
x1=C;
hence thesis by A59,ENUMSET1:def 1;
end;
case
x1=D;
hence thesis by A59,ENUMSET1:def 1;
end;
end;
hence thesis;
end;
for y being set holds y in rng h implies m in y
proof
let y be set;
assume
A60: y in rng h;
now
per cases by A57,A60,ENUMSET1:def 1;
case
y=h.B;
hence thesis by A56,XBOOLE_0:def 4;
end;
case
y=h.C;
hence thesis by A56,XBOOLE_0:def 4;
end;
case
y=h.D;
hence thesis by A55,XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then m in meet (rng h) by A45,SETFAM_1:def 1;
hence thesis by A40,A42,A45,SETFAM_1:def 9;
end;
then (h.B /\ h.C) /\ h.D = x by A47,XBOOLE_0:def 10;
then x in INTERSECTION(B '/\' C,D) by A51,A54,SETFAM_1:def 5;
then x in INTERSECTION(B '/\' C,D) \ {{}} by A52,XBOOLE_0:def 5;
hence thesis by PARTIT1:def 4;
end;
then '/\' (G \ {A}) = B '/\' C '/\' D by A13,XBOOLE_0:def 10;
hence thesis by BVFUNC_2:def 7;
end;
end;
theorem Th8:
G={A,B,C,D} & A<>B & B<>C & B<>D implies CompF(B,G) = A '/\' C '/\' D
proof
{A,B,C,D} = {B,A,C,D} by ENUMSET1:65;
hence thesis by Th7;
end;
theorem
G={A,B,C,D} & A<>C & B<>C & C<>D implies CompF(C,G) = A '/\' B '/\' D
proof
{A,B,C,D} = {C,A,B,D} by ENUMSET1:67;
hence thesis by Th7;
end;
theorem
G={A,B,C,D} & A<>D & B<>D & C<>D implies CompF(D,G) = A '/\' C '/\' B
proof
{A,B,C,D} = {D,A,C,B} by ENUMSET1:70;
hence thesis by Th7;
end;
theorem
for B,C,D,b,c,d being object
holds dom (B,C,D) --> (b,c,d) = {B,C,D}by FUNCT_4:128;
theorem
for f being Function, C,D,c,d being object st C<>D holds (f +* (C .--> c)
+* (D .--> d)).C = c by Lm1;
theorem
for B,C,D,b,c,d being object st B<>C & D<>B holds
((B,C,D) --> (b,c,d)).B = b by FUNCT_4:134;
theorem
for B,C,D,b,c,d being object, h being Function
st h = (B,C,D) --> (b,c,d)
holds rng h = {h.B,h.C,h.D} by Lm2;
:: from BVFUNC20
theorem Th15:
for h being Function, A9,B9,C9,D9 being object st A<>B & A<>C & A<>
D & B<>C & B<>D & C<>D & h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (A
.--> A9) holds h.B = B9 & h.C = C9 & h.D = D9
proof
let h be Function;
let A9,B9,C9,D9 be object;
assume that
A1: A<>B and
A2: A<>C and
A3: A<>D and
A4: B<>C and
A5: B<>D and
A6: C<>D and
A7: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (A .--> A9);
not D in dom (A .--> A9) by A3,TARSKI:def 1;
then
A9: h.D=((B .--> B9) +* (C .--> C9) +* (D .--> D9)).D by A7,FUNCT_4:11;
not C in dom (A .--> A9) by A2,TARSKI:def 1;
then
A10: h.C=((B .--> B9) +* (C .--> C9) +* (D .--> D9)).C by A7,FUNCT_4:11;
not C in dom (D .--> D9) by A6,TARSKI:def 1;
then
A12: h.C=((B .--> B9) +* (C .--> C9)).C by A10,FUNCT_4:11;
not B in dom (A .--> A9) by A1,TARSKI:def 1;
then
A13: h.B=((B .--> B9) +* (C .--> C9) +* (D .--> D9)).B by A7,FUNCT_4:11;
not B in dom (D .--> D9) by A5,TARSKI:def 1;
then
A14: h.B=((B .--> B9) +* (C .--> C9)).B by A13,FUNCT_4:11;
not B in dom (C .--> C9) by A4,TARSKI:def 1;
then h.B=(B .--> B9).B by A14,FUNCT_4:11;
hence h.B = B9 by FUNCOP_1:72;
C in dom (C .--> C9) by TARSKI:def 1;
then h.C=(C .--> C9).C by A12,FUNCT_4:13;
hence h.C = C9 by FUNCOP_1:72;
D in dom (D .--> D9) by TARSKI:def 1;
then h.D=(D .--> D9).D by A9,FUNCT_4:13;
hence thesis by FUNCOP_1:72;
end;
theorem Th16:
for A,B,C,D being object,h being Function,
A9,B9,C9,D9 being object st
h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (A .--> A9) holds dom h = {A,B
,C,D}
proof
let A,B,C,D be object;
let h be Function;
let A9,B9,C9,D9 be object;
assume
A1: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (A .--> A9);
dom ((B .--> B9) +* (C .--> C9)) = dom (B .--> B9) \/ dom (C .--> C9) by
FUNCT_4:def 1;
then
A2: dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9)) = dom (B .--> B9) \/ dom
(C .--> C9) \/ dom (D .--> D9) by FUNCT_4:def 1;
dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (A .--> A9)) = {B} \/
dom (C .--> C9) \/ dom (D .--> D9) \/ dom (A .--> A9) by A2,FUNCT_4:def 1
.= {B} \/ {C} \/ dom (D .--> D9) \/ dom (A .--> A9)
.= {B} \/ {C} \/ {D} \/ dom (A .--> A9)
.= {A} \/ (({B} \/ {C}) \/ {D})
.= {A} \/ ({B,C} \/ {D}) by ENUMSET1:1
.= {A} \/ {B,C,D} by ENUMSET1:3
.= {A,B,C,D} by ENUMSET1:4;
hence thesis by A1;
end;
theorem Th17:
for h being Function,A9,B9,C9,D9 being object st G={A,B,C,D} & h =
(B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (A .--> A9) holds rng h = {h.A,h.B
,h.C,h.D}
proof
let h be Function;
let A9,B9,C9,D9 be object;
assume that
A1: G={A,B,C,D} and
A2: h = (B .--> B9)+*(C .--> C9)+*(D .--> D9)+*(A .--> A9);
A3: dom h = G by A1,A2,Th16;
then
A4: B in dom h by A1,ENUMSET1:def 2;
A5: rng h c= {h.A,h.B,h.C,h.D}
proof
let t be object;
assume t in rng h;
then consider x1 being object such that
A6: x1 in dom h and
A7: t = h.x1 by FUNCT_1:def 3;
now
per cases by A1,A3,A6,ENUMSET1:def 2;
case
x1=A;
hence thesis by A7,ENUMSET1:def 2;
end;
case
x1=B;
hence thesis by A7,ENUMSET1:def 2;
end;
case
x1=C;
hence thesis by A7,ENUMSET1:def 2;
end;
case
x1=D;
hence thesis by A7,ENUMSET1:def 2;
end;
end;
hence thesis;
end;
A8: D in dom h by A1,A3,ENUMSET1:def 2;
A9: C in dom h by A1,A3,ENUMSET1:def 2;
A10: A in dom h by A1,A3,ENUMSET1:def 2;
{h.A,h.B,h.C,h.D} c= rng h
proof
let t be object;
assume
A11: t in {h.A,h.B,h.C,h.D};
per cases by A11,ENUMSET1:def 2;
suppose
t=h.A;
hence thesis by A10,FUNCT_1:def 3;
end;
suppose
t=h.B;
hence thesis by A4,FUNCT_1:def 3;
end;
suppose
t=h.C;
hence thesis by A9,FUNCT_1:def 3;
end;
suppose
t=h.D;
hence thesis by A8,FUNCT_1:def 3;
end;
end;
hence thesis by A5,XBOOLE_0:def 10;
end;
theorem
for z,u being Element of Y, h being Function st G is independent & G={
A,B,C,D} & A<>B & A<>C & A<>D & B<>C & B<>D & C<>D holds EqClass(u,B '/\' C
'/\' D) meets EqClass(z,A)
proof
let z,u be Element of Y;
let h be Function;
assume that
A1: G is independent and
A2: G={A,B,C,D} and
A3: A<>B & A<>C & A<>D & B<>C & B<>D & C<>D;
set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (D .--> EqClass(u,
D)) +* (A .--> EqClass(z,A));
A4: h.B = EqClass(u,B) by A3,Th15;
A5: h.D = EqClass(u,D) by A3,Th15;
A6: h.C = EqClass(u,C) by A3,Th15;
A7: rng h = {h.A,h.B,h.C,h.D} by A2,Th17;
rng h c= bool Y
proof
let t be object;
assume
A8: t in rng h;
per cases by A7,A8,ENUMSET1:def 2;
suppose
t=h.A;
then t=EqClass(z,A) by FUNCT_7:94;
hence thesis;
end;
suppose
t=h.B;
hence thesis by A4;
end;
suppose
t=h.C;
hence thesis by A6;
end;
suppose
t=h.D;
hence thesis by A5;
end;
end;
then reconsider FF=rng h as Subset-Family of Y;
A9: dom h = G by A2,Th16;
for d being set st d in G holds h.d in d
proof
let d be set;
assume
A10: d in G;
per cases by A2,A10,ENUMSET1:def 2;
suppose
A11: d=A;
h.A=EqClass(z,A) by FUNCT_7:94;
hence thesis by A11;
end;
suppose
A12: d=B;
h.B=EqClass(u,B) by A3,Th15;
hence thesis by A12;
end;
suppose
A13: d=C;
h.C=EqClass(u,C) by A3,Th15;
hence thesis by A13;
end;
suppose
A14: d=D;
h.D=EqClass(u,D) by A3,Th15;
hence thesis by A14;
end;
end;
then Intersect FF <>{} by A1,A9,BVFUNC_2:def 5;
then consider m being object such that
A15: m in Intersect FF by XBOOLE_0:def 1;
A in dom h by A2,A9,ENUMSET1:def 2;
then
A16: h.A in rng h by FUNCT_1:def 3;
then
A17: m in meet FF by A15,SETFAM_1:def 9;
D in dom h by A2,A9,ENUMSET1:def 2;
then h.D in rng h by FUNCT_1:def 3;
then
A18: m in h.D by A17,SETFAM_1:def 1;
C in dom h by A2,A9,ENUMSET1:def 2;
then h.C in rng h by FUNCT_1:def 3;
then
A19: m in h.C by A17,SETFAM_1:def 1;
B in dom h by A2,A9,ENUMSET1:def 2;
then h.B in rng h by FUNCT_1:def 3;
then m in h.B by A17,SETFAM_1:def 1;
then m in EqClass(u,B) /\ EqClass(u,C) by A4,A6,A19,XBOOLE_0:def 4;
then
A20: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A5,A18,XBOOLE_0:def 4
;
h.A = EqClass(z,A) & m in h.A by A16,A17,FUNCT_7:94,SETFAM_1:def 1;
then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(z,A) by A20
,XBOOLE_0:def 4;
then
A21: EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) meets EqClass(z,A) by
XBOOLE_0:4;
EqClass(u,B '/\' C '/\' D) = EqClass(u,B '/\' C) /\ EqClass(u,D) by Th1;
hence thesis by A21,Th1;
end;
theorem
for z,u being Element of Y st G is independent & G={A,B,C,D} & A<>B &
A<>C & A<>D & B<>C & B<>D & C<>D & EqClass(z,C '/\' D)=EqClass(u,C '/\' D)
holds EqClass(u,CompF(A,G)) meets EqClass(z,CompF(B,G))
proof
let z,u be Element of Y;
assume that
A1: G is independent and
A2: G={A,B,C,D} and
A3: A<>B and
A4: A<>C & A<>D and
A5: B<>C & B<>D and
A6: C<>D and
A7: EqClass(z,C '/\' D)=EqClass(u,C '/\' D);
set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (D .--> EqClass(u,
D)) +* (A .--> EqClass(z,A));
set H=EqClass(z,CompF(B,G));
A8: A '/\' (C '/\' D) = A '/\' C '/\' D by PARTIT1:14;
A9: rng h = {h.A,h.B,h.C,h.D} by A2,Th17;
rng h c= bool Y
proof
let t be object;
assume
A10: t in rng h;
per cases by A9,A10,ENUMSET1:def 2;
suppose
t=h.A;
then t=EqClass(z,A) by FUNCT_7:94;
hence thesis;
end;
suppose
t=h.B;
then t=EqClass(u,B) by A3,A4,A5,A6,Th15;
hence thesis;
end;
suppose
t=h.C;
then t=EqClass(u,C) by A3,A4,A5,A6,Th15;
hence thesis;
end;
suppose
t=h.D;
then t=EqClass(u,D) by A3,A4,A5,A6,Th15;
hence thesis;
end;
end;
then reconsider FF=rng h as Subset-Family of Y;
set I=EqClass(z,A), GG=EqClass(u,B '/\' C '/\' D);
A11: GG = EqClass(u,B '/\' C) /\ EqClass(u,D) by Th1;
A12: for d being set st d in G holds h.d in d
proof
let d be set;
assume
A13: d in G;
per cases by A2,A13,ENUMSET1:def 2;
suppose
A14: d=A;
h.A=EqClass(z,A) by FUNCT_7:94;
hence thesis by A14;
end;
suppose
A15: d=B;
h.B=EqClass(u,B) by A3,A4,A5,A6,Th15;
hence thesis by A15;
end;
suppose
A16: d=C;
h.C=EqClass(u,C) by A3,A4,A5,A6,Th15;
hence thesis by A16;
end;
suppose
A17: d=D;
h.D=EqClass(u,D) by A3,A4,A5,A6,Th15;
hence thesis by A17;
end;
end;
dom h=G by A2,Th16;
then (Intersect FF)<>{} by A1,A12,BVFUNC_2:def 5;
then consider m being object such that
A18: m in Intersect FF by XBOOLE_0:def 1;
A19: dom h = G by A2,Th16;
then A in dom h by A2,ENUMSET1:def 2;
then
A20: h.A in rng h by FUNCT_1:def 3;
then
A21: m in meet FF by A18,SETFAM_1:def 9;
then
A22: h.A = EqClass(z,A) & m in h.A by A20,FUNCT_7:94,SETFAM_1:def 1;
D in dom h by A2,A19,ENUMSET1:def 2;
then h.D in rng h by FUNCT_1:def 3;
then
A23: m in h.D by A21,SETFAM_1:def 1;
C in dom h by A2,A19,ENUMSET1:def 2;
then h.C in rng h by FUNCT_1:def 3;
then
A24: m in h.C by A21,SETFAM_1:def 1;
B in dom h by A2,A19,ENUMSET1:def 2;
then h.B in rng h by FUNCT_1:def 3;
then
A25: m in h.B by A21,SETFAM_1:def 1;
h.B = EqClass(u,B) & h.C = EqClass(u,C) by A3,A4,A5,A6,Th15;
then
A26: m in EqClass(u,B) /\ EqClass(u,C) by A25,A24,XBOOLE_0:def 4;
h.D = EqClass(u,D) by A3,A4,A5,A6,Th15;
then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A23,A26,
XBOOLE_0:def 4;
then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(z,A) by A22
,XBOOLE_0:def 4;
then GG /\ I <> {} by A11,Th1;
then consider p being object such that
A27: p in GG /\ I by XBOOLE_0:def 1;
reconsider p as Element of Y by A27;
set K=EqClass(p,C '/\' D);
A28: p in GG by A27,XBOOLE_0:def 4;
set L=EqClass(z,C '/\' D);
A29: z in I by EQREL_1:def 6;
GG = EqClass(u,B '/\' (C '/\' D)) by PARTIT1:14;
then
A30: GG c= EqClass(u,C '/\' D) by BVFUNC11:3;
p in EqClass(p,C '/\' D) by EQREL_1:def 6;
then K meets L by A7,A30,A28,XBOOLE_0:3;
then K=L by EQREL_1:41;
then z in K by EQREL_1:def 6;
then
A31: z in I /\ K by A29,XBOOLE_0:def 4;
A32: p in K & p in I by A27,EQREL_1:def 6,XBOOLE_0:def 4;
then p in I /\ K by XBOOLE_0:def 4;
then I /\ K in INTERSECTION(A,C '/\' D) & not I /\ K in {{}} by
SETFAM_1:def 5,TARSKI:def 1;
then
A33: I /\ K in INTERSECTION(A,C '/\' D) \ {{}} by XBOOLE_0:def 5;
CompF(B,G) = A '/\' C '/\' D by A2,A3,A5,Th8;
then I /\ K in CompF(B,G) by A33,A8,PARTIT1:def 4;
then
A34: I /\ K = H or I /\ K misses H by EQREL_1:def 4;
z in H by EQREL_1:def 6;
then p in H by A32,A31,A34,XBOOLE_0:3,def 4;
then p in GG /\ H by A28,XBOOLE_0:def 4;
then GG meets H by XBOOLE_0:4;
hence thesis by A2,A3,A4,Th7;
end;
theorem
for z,u being Element of Y st G is independent & G={A,B,C} & A<>B & B
<>C & C<>A & EqClass(z,C)=EqClass(u,C) holds EqClass(u,CompF(A,G)) meets
EqClass(z,CompF(B,G))
proof
let z,u be Element of Y;
assume that
A1: G is independent and
A2: G={A,B,C} and
A3: A<>B and
A4: B<>C and
A5: C<>A and
A6: EqClass(z,C)=EqClass(u,C);
set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (A .--> EqClass(z,
A));
A in dom (A .--> EqClass(z,A)) by TARSKI:def 1;
then h.A = (A .--> EqClass(z,A)).A by FUNCT_4:13;
then
A8: h.A = EqClass(z,A) by FUNCOP_1:72;
set H=EqClass(z,CompF(B,G)), I=EqClass(z,A), GG=EqClass(u,B '/\' C);
A9: GG /\ I = EqClass(u,B) /\ EqClass(u,C) /\ EqClass(z,A) by Th1;
dom ((B .--> EqClass(u,B)) +* (C .--> EqClass(u,C))) = dom (B .-->
EqClass(u,B)) \/ dom (C .--> EqClass(u,C)) by FUNCT_4:def 1;
then
A10: dom ((B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (A .--> EqClass(
z,A))) = dom (B .--> EqClass(u,B)) \/ dom (C .--> EqClass(u,C)) \/ dom (A .-->
EqClass(z,A)) by FUNCT_4:def 1;
A12: C in dom (C .--> EqClass(u,C)) by TARSKI:def 1;
not B in dom (A .--> EqClass(z,A)) by A3,TARSKI:def 1;
then
A13: h.B=((B .--> EqClass(u,B)) +* (C .--> EqClass(u,C))).B by FUNCT_4:11;
not B in dom (C .--> EqClass(u,C)) by A4,TARSKI:def 1;
then h.B=(B .--> EqClass(u,B)).B by A13,FUNCT_4:11;
then
A14: h.B = EqClass(u,B) by FUNCOP_1:72;
not C in dom (A .--> EqClass(z,A)) by A5,TARSKI:def 1;
then h.C=((B .--> EqClass(u,B)) +* (C .--> EqClass(u,C))).C by FUNCT_4:11;
then h.C=(C .--> EqClass(u,C)).C by A12,FUNCT_4:13;
then
A15: h.C = EqClass(u,C) by FUNCOP_1:72;
A16: dom ((B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (A .--> EqClass(
z,A))) = {A} \/ {B} \/ {C} by A10,XBOOLE_1:4
.= {A,B} \/ {C} by ENUMSET1:1
.= {A,B,C} by ENUMSET1:3;
A17: rng h c= {h.A,h.B,h.C}
proof
let t be object;
assume t in rng h;
then consider x1 being object such that
A18: x1 in dom h and
A19: t = h.x1 by FUNCT_1:def 3;
now
per cases by A16,A18,ENUMSET1:def 1;
case
x1=A;
hence thesis by A19,ENUMSET1:def 1;
end;
case
x1=B;
hence thesis by A19,ENUMSET1:def 1;
end;
case
x1=C;
hence thesis by A19,ENUMSET1:def 1;
end;
end;
hence thesis;
end;
rng h c= bool Y
proof
let t be object;
assume
A20: t in rng h;
now
per cases by A17,A20,ENUMSET1:def 1;
case
t=h.A;
hence thesis by A8;
end;
case
t=h.B;
hence thesis by A14;
end;
case
t=h.C;
hence thesis by A15;
end;
end;
hence thesis;
end;
then reconsider FF=rng h as Subset-Family of Y;
A21: z in H by EQREL_1:def 6;
for d being set st d in G holds h.d in d
proof
let d be set;
assume
A22: d in G;
now
per cases by A2,A22,ENUMSET1:def 1;
case
d=A;
hence thesis by A8;
end;
case
d=B;
hence thesis by A14;
end;
case
d=C;
hence thesis by A15;
end;
end;
hence thesis;
end;
then (Intersect FF)<>{} by A1,A2,A16,BVFUNC_2:def 5;
then consider m being object such that
A23: m in Intersect FF by XBOOLE_0:def 1;
A in dom h by A16,ENUMSET1:def 1;
then
A24: h.A in rng h by FUNCT_1:def 3;
then
A25: Intersect FF = meet (rng h) by SETFAM_1:def 9;
C in dom h by A16,ENUMSET1:def 1;
then h.C in rng h by FUNCT_1:def 3;
then
A26: m in h.C by A25,A23,SETFAM_1:def 1;
B in dom h by A16,ENUMSET1:def 1;
then h.B in rng h by FUNCT_1:def 3;
then m in h.B by A25,A23,SETFAM_1:def 1;
then
A27: m in EqClass(u,B) /\ EqClass(u,C) by A14,A15,A26,XBOOLE_0:def 4;
m in h.A by A24,A25,A23,SETFAM_1:def 1;
then GG /\ I <> {} by A8,A9,A27,XBOOLE_0:def 4;
then consider p being object such that
A28: p in GG /\ I by XBOOLE_0:def 1;
reconsider p as Element of Y by A28;
set K=EqClass(p,C);
A29: I /\ K in INTERSECTION(A,C) by SETFAM_1:def 5;
set L=EqClass(z,C);
A30: p in EqClass(p,C) by EQREL_1:def 6;
A31: p in GG by A28,XBOOLE_0:def 4;
p in K & p in I by A28,EQREL_1:def 6,XBOOLE_0:def 4;
then
A32: p in I /\ K by XBOOLE_0:def 4;
then not I /\ K in {{}} by TARSKI:def 1;
then I /\ K in INTERSECTION(A,C) \ {{}} by A29,XBOOLE_0:def 5;
then
A33: I /\ K in A '/\' C by PARTIT1:def 4;
GG c= L by A6,BVFUNC11:3;
then K meets L by A31,A30,XBOOLE_0:3;
then K=L by EQREL_1:41;
then
A34: z in K by EQREL_1:def 6;
z in I by EQREL_1:def 6;
then
A35: z in I /\ K by A34,XBOOLE_0:def 4;
CompF(B,G) = A '/\' C by A2,A3,A4,Th5;
then
A36: I /\ K = H or I /\ K misses H by A33,EQREL_1:def 4;
GG=EqClass(u,CompF(A,G)) by A2,A3,A5,Th4;
hence thesis by A32,A31,A35,A21,A36,XBOOLE_0:3;
end;
theorem Th21:
G={A,B,C,D,E} & A<>B & A<>C & A<>D & A<>E implies CompF(A,G) = B
'/\' C '/\' D '/\' E
proof
assume that
A1: G={A,B,C,D,E} and
A2: A<>B and
A3: A<>C and
A4: A<>D and
A5: A<>E;
per cases;
suppose
A6: B = C;
then G = {A,B,B,D} \/ {E} by A1,ENUMSET1:10
.= {B,B,A,D} \/ {E} by ENUMSET1:67
.= {B,B,A,D,E} by ENUMSET1:10
.= {B,A,D,E} by ENUMSET1:32
.={A,B,D,E} by ENUMSET1:65;
hence CompF(A,G) = B '/\' D '/\' E by A2,A4,A5,Th7
.= B '/\' C '/\' D '/\' E by A6,PARTIT1:13;
end;
suppose
A7: B = D;
then G = {A,B,C,B} \/ {E} by A1,ENUMSET1:10
.= {B,B,A,C} \/ {E} by ENUMSET1:69
.= {B,B,A,C,E} by ENUMSET1:10
.= {B,A,C,E} by ENUMSET1:32
.={A,B,C,E} by ENUMSET1:65;
hence CompF(A,G) = B '/\' C '/\' E by A2,A3,A5,Th7
.= B '/\' D '/\' C '/\' E by A7,PARTIT1:13
.= B '/\' C '/\' D '/\' E by PARTIT1:14;
end;
suppose
A8: B = E;
then G = {A} \/ {B,C,D,B} by A1,ENUMSET1:7
.= {A} \/ {B,B,C,D} by ENUMSET1:63
.= {A} \/ {B,C,D} by ENUMSET1:31
.={A,B,C,D} by ENUMSET1:4;
hence CompF(A,G) = B '/\' C '/\' D by A2,A3,A4,Th7
.= B '/\' E '/\' C '/\' D by A8,PARTIT1:13
.= B '/\' E '/\' (C '/\' D) by PARTIT1:14
.= B '/\' (C '/\' D) '/\' E by PARTIT1:14
.= B '/\' C '/\' D '/\' E by PARTIT1:14;
end;
suppose
A9: C = D;
then G = {A,B,C,C} \/ {E} by A1,ENUMSET1:10
.= {C,C,A,B} \/ {E} by ENUMSET1:73
.= {C,A,B} \/ {E} by ENUMSET1:31
.= {C,A,B,E} by ENUMSET1:6
.={A,B,C,E} by ENUMSET1:67;
hence CompF(A,G) = B '/\' C '/\' E by A2,A3,A5,Th7
.= B '/\' (C '/\' D) '/\' E by A9,PARTIT1:13
.= B '/\' C '/\' D '/\' E by PARTIT1:14;
end;
suppose
A10: C = E;
then G = {A} \/ {B,C,D,C} by A1,ENUMSET1:7
.= {A} \/ {C,C,B,D} by ENUMSET1:72
.= {A} \/ {C,B,D} by ENUMSET1:31
.= {A,C,B,D} by ENUMSET1:4
.={A,B,C,D} by ENUMSET1:62;
hence CompF(A,G) = B '/\' C '/\' D by A2,A3,A4,Th7
.= B '/\' (C '/\' E) '/\' D by A10,PARTIT1:13
.= B '/\' (C '/\' E '/\' D) by PARTIT1:14
.= B '/\' (C '/\' D '/\' E) by PARTIT1:14
.= B '/\' (C '/\' D) '/\' E by PARTIT1:14
.= B '/\' C '/\' D '/\' E by PARTIT1:14;
end;
suppose
A11: D = E;
then G = {A} \/ {B,C,D,D} by A1,ENUMSET1:7
.= {A} \/ {D,D,B,C} by ENUMSET1:73
.= {A} \/ {D,B,C} by ENUMSET1:31
.= {A,D,B,C} by ENUMSET1:4
.={A,B,C,D} by ENUMSET1:63;
hence CompF(A,G) = B '/\' C '/\' D by A2,A3,A4,Th7
.= B '/\' C '/\' (D '/\' E) by A11,PARTIT1:13
.= B '/\' (C '/\' (D '/\' E)) by PARTIT1:14
.= B '/\' (C '/\' D '/\' E) by PARTIT1:14
.= B '/\' (C '/\' D) '/\' E by PARTIT1:14
.= B '/\' C '/\' D '/\' E by PARTIT1:14;
end;
suppose
A12: B<>C & B<>D & B<>E & C<>D & C<>E & D<>E;
A13: ( not D in {A})& not E in {A} by A4,A5,TARSKI:def 1;
A14: not B in {A} by A2,TARSKI:def 1;
G \ {A}={A} \/ {B,C,D,E} \ {A} by A1,ENUMSET1:7;
then
A15: G \ {A} = ({A} \ {A}) \/ ({B,C,D,E} \ {A}) by XBOOLE_1:42;
A16: not C in {A} by A3,TARSKI:def 1;
A in {A} by TARSKI:def 1;
then
A17: {A} \ {A}={} by ZFMISC_1:60;
{B,C,D,E} \ {A} = ({B} \/ {C,D,E}) \ {A} by ENUMSET1:4
.= ({B} \ {A}) \/ ({C,D,E} \ {A}) by XBOOLE_1:42
.= {B} \/ ({C,D,E} \ {A}) by A14,ZFMISC_1:59
.= {B} \/ (({C} \/ {D,E}) \ {A}) by ENUMSET1:2
.= {B} \/ (({C} \ {A}) \/ ({D,E} \ {A})) by XBOOLE_1:42
.= {B} \/ (({C} \ {A}) \/ {D,E}) by A13,ZFMISC_1:63
.= {B} \/ ({C} \/ {D,E}) by A16,ZFMISC_1:59
.= {B} \/ {C,D,E} by ENUMSET1:2;
then
A18: G \ {A} = ({A} \ {A}) \/ {B,C,D,E} by A15,ENUMSET1:4;
A19: B '/\' C '/\' D '/\' E c= '/\' (G \ {A})
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume
A20: x in B '/\' C '/\' D '/\' E;
then
A21: x<>{} by EQREL_1:def 4;
x in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A20,PARTIT1:def 4;
then consider bcd,e being set such that
A22: bcd in B '/\' C '/\' D and
A23: e in E and
A24: x = bcd /\ e by SETFAM_1:def 5;
bcd in INTERSECTION(B '/\' C,D) \ {{}} by A22,PARTIT1:def 4;
then consider bc,d being set such that
A25: bc in B '/\' C and
A26: d in D and
A27: bcd = bc /\ d by SETFAM_1:def 5;
bc in INTERSECTION(B,C) \ {{}} by A25,PARTIT1:def 4;
then consider b,c being set such that
A28: b in B and
A29: c in C and
A30: bc = b /\ c by SETFAM_1:def 5;
set h = (B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e);
A32: C in dom (C .--> c) by TARSKI:def 1;
A34: D in dom (D .--> d) by TARSKI:def 1;
A35: not C in dom (D .--> d) by A12,TARSKI:def 1;
E in dom (E .--> e) by TARSKI:def 1;
then
A37: h.E = (E .--> e).E by FUNCT_4:13;
then
A38: h.E = e by FUNCOP_1:72;
not C in dom (E .--> e) by A12,TARSKI:def 1;
then h.C=((B .--> b) +* (C .--> c) +* (D .--> d)).C by FUNCT_4:11;
then h.C=((B .--> b) +* (C .--> c)).C by A35,FUNCT_4:11;
then
A39: h.C=(C .--> c).C by A32,FUNCT_4:13;
then
A40: h.C = c by FUNCOP_1:72;
not D in dom (E .--> e) by A12,TARSKI:def 1;
then h.D=((B .--> b) +* (C .--> c) +* (D .--> d)).D by FUNCT_4:11;
then
A41: h.D = (D .--> d).D by A34,FUNCT_4:13;
then
A42: h.D = d by FUNCOP_1:72;
A43: not B in dom (C .--> c) by A12,TARSKI:def 1;
A44: not B in dom (D .--> d) by A12,TARSKI:def 1;
not B in dom (E .--> e) by A12,TARSKI:def 1;
then h.B=((B .--> b) +* (C .--> c) +* (D .--> d)).B by FUNCT_4:11;
then h.B=((B .--> b) +* (C .--> c)).B by A44,FUNCT_4:11;
then
A45: h.B=(B .--> b).B by A43,FUNCT_4:11;
then
A46: h.B = b by FUNCOP_1:72;
A47: for p being set st p in (G \ {A}) holds h.p in p
proof
let p be set;
assume
A48: p in (G \ {A});
now
per cases by A15,A17,A48,ENUMSET1:def 2;
case
p=D;
hence thesis by A26,A41,FUNCOP_1:72;
end;
case
p=B;
hence thesis by A28,A45,FUNCOP_1:72;
end;
case
p=C;
hence thesis by A29,A39,FUNCOP_1:72;
end;
case
p=E;
hence thesis by A23,A37,FUNCOP_1:72;
end;
end;
hence thesis;
end;
dom ((B .--> b) +* (C .--> c)) = dom (B .--> b) \/ dom (C .--> c)
by FUNCT_4:def 1;
then dom ((B .--> b) +* (C .--> c) +* (D .--> d)) = dom (B .--> b) \/
dom (C .--> c) \/ dom (D .--> d) by FUNCT_4:def 1;
then
A49: dom ((B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e)) = dom (B
.--> b) \/ dom (C .--> c) \/ dom (D .--> d) \/ dom (E .--> e) by
FUNCT_4:def 1;
A50: dom ((B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e)) = {B} \/
{C} \/ {D} \/ {E} by A49
.= {B,C} \/ {D} \/ {E} by ENUMSET1:1
.= {B,C,D} \/ {E} by ENUMSET1:3
.= {B,C,D,E} by ENUMSET1:6;
then
A51: D in dom h by ENUMSET1:def 2;
A52: rng h c= {h.D,h.B,h.C,h.E}
proof
let t be object;
assume t in rng h;
then consider x1 being object such that
A53: x1 in dom h and
A54: t = h.x1 by FUNCT_1:def 3;
now
per cases by A50,A53,ENUMSET1:def 2;
case
x1=D;
hence thesis by A54,ENUMSET1:def 2;
end;
case
x1=B;
hence thesis by A54,ENUMSET1:def 2;
end;
case
x1=C;
hence thesis by A54,ENUMSET1:def 2;
end;
case
x1=E;
hence thesis by A54,ENUMSET1:def 2;
end;
end;
hence thesis;
end;
rng h c= bool Y
proof
let t be object;
assume
A55: t in rng h;
now
per cases by A52,A55,ENUMSET1:def 2;
case
t=h.D;
hence thesis by A26,A42;
end;
case
t=h.B;
hence thesis by A28,A46;
end;
case
t=h.C;
hence thesis by A29,A40;
end;
case
t=h.E;
hence thesis by A23,A38;
end;
end;
hence thesis;
end;
then reconsider F=rng h as Subset-Family of Y;
A56: C in dom h by A50,ENUMSET1:def 2;
A57: E in dom h by A50,ENUMSET1:def 2;
A58: B in dom h by A50,ENUMSET1:def 2;
A59: {h.D,h.B,h.C,h.E} c= rng h
proof
let t be object;
assume
A60: t in {h.D,h.B,h.C,h.E};
now
per cases by A60,ENUMSET1:def 2;
case
t=h.D;
hence thesis by A51,FUNCT_1:def 3;
end;
case
t=h.B;
hence thesis by A58,FUNCT_1:def 3;
end;
case
t=h.C;
hence thesis by A56,FUNCT_1:def 3;
end;
case
t=h.E;
hence thesis by A57,FUNCT_1:def 3;
end;
end;
hence thesis;
end;
then
A61: {h.D,h.B,h.C,h.E} = rng h by A52,XBOOLE_0:def 10;
reconsider h as Function;
A62: xx c= Intersect F
proof
let u be object;
A63: h.D in {h.D,h.B,h.C,h.E} by ENUMSET1:def 2;
assume
A64: u in xx;
for y be set holds y in F implies u in y
proof
let y be set;
assume
A65: y in F;
now
per cases by A52,A65,ENUMSET1:def 2;
case
A66: y=h.D;
u in d /\ ((b /\ c) /\ e) by A24,A27,A30,A64,XBOOLE_1:16;
hence thesis by A42,A66,XBOOLE_0:def 4;
end;
case
A67: y=h.B;
u in (c /\ (d /\ b)) /\ e by A24,A27,A30,A64,XBOOLE_1:16;
then u in c /\ ((d /\ b) /\ e) by XBOOLE_1:16;
then u in c /\ ((d /\ e) /\ b) by XBOOLE_1:16;
then u in (c /\ (d /\ e)) /\ b by XBOOLE_1:16;
hence thesis by A46,A67,XBOOLE_0:def 4;
end;
case
A68: y=h.C;
u in (c /\ (b /\ d)) /\ e by A24,A27,A30,A64,XBOOLE_1:16;
then u in c /\ (b /\ d /\ e) by XBOOLE_1:16;
hence thesis by A40,A68,XBOOLE_0:def 4;
end;
case
y=h.E;
hence thesis by A24,A38,A64,XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u in meet F by A59,A63,SETFAM_1:def 1;
hence thesis by A59,A63,SETFAM_1:def 9;
end;
A69: rng h <> {} by A51,FUNCT_1:3;
Intersect F c= xx
proof
let t be object;
assume t in Intersect F;
then
A70: t in meet (rng h) by A69,SETFAM_1:def 9;
h.D in rng h by A61,ENUMSET1:def 2;
then
A71: t in h.D by A70,SETFAM_1:def 1;
h.C in rng h by A61,ENUMSET1:def 2;
then
A72: t in h.C by A70,SETFAM_1:def 1;
h.B in rng h by A61,ENUMSET1:def 2;
then t in h.B by A70,SETFAM_1:def 1;
then t in b /\ c by A46,A40,A72,XBOOLE_0:def 4;
then
A73: t in (b /\ c) /\ d by A42,A71,XBOOLE_0:def 4;
h.E in rng h by A61,ENUMSET1:def 2;
then t in h.E by A70,SETFAM_1:def 1;
hence thesis by A24,A27,A30,A38,A73,XBOOLE_0:def 4;
end;
then x = Intersect F by A62,XBOOLE_0:def 10;
hence thesis by A18,A17,A50,A47,A21,BVFUNC_2:def 1;
end;
'/\' (G \ {A}) c= B '/\' C '/\' D '/\' E
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume x in '/\' (G \ {A});
then consider h being Function, F being Subset-Family of Y such that
A74: dom h=(G \ {A}) and
A75: rng h = F and
A76: for d being set st d in (G \ {A}) holds h.d in d and
A77: x=Intersect F and
A78: x<>{} by BVFUNC_2:def 1;
D in dom h by A18,A17,A74,ENUMSET1:def 2;
then
A79: h.D in rng h by FUNCT_1:def 3;
set mbc=h.B /\ h.C;
A80: not x in {{}} by A78,TARSKI:def 1;
E in (G \ {A}) by A18,A17,ENUMSET1:def 2;
then
A81: h.E in E by A76;
D in (G \ {A}) by A18,A17,ENUMSET1:def 2;
then
A82: h.D in D by A76;
C in (G \ {A}) by A18,A17,ENUMSET1:def 2;
then
A83: h.C in C by A76;
E in dom h by A18,A17,A74,ENUMSET1:def 2;
then
A84: h.E in rng h by FUNCT_1:def 3;
set mbcd=(h.B /\ h.C) /\ h.D;
B in dom h by A18,A17,A74,ENUMSET1:def 2;
then
A85: h.B in rng h by FUNCT_1:def 3;
C in dom h by A18,A17,A74,ENUMSET1:def 2;
then
A86: h.C in rng h by FUNCT_1:def 3;
A87: xx c= h.B /\ h.C /\ h.D /\ h.E
proof
let m be object;
assume m in xx;
then
A88: m in meet (rng h) by A75,A77,A85,SETFAM_1:def 9;
then m in h.B & m in h.C by A85,A86,SETFAM_1:def 1;
then
A89: m in h.B /\ h.C by XBOOLE_0:def 4;
m in h.D by A79,A88,SETFAM_1:def 1;
then
A90: m in h.B /\ h.C /\ h.D by A89,XBOOLE_0:def 4;
m in h.E by A84,A88,SETFAM_1:def 1;
hence thesis by A90,XBOOLE_0:def 4;
end;
then mbcd<>{} by A78;
then
A91: not mbcd in {{}} by TARSKI:def 1;
mbc<>{} by A78,A87;
then
A92: not mbc in {{}} by TARSKI:def 1;
B in (G \ {A}) by A18,A17,ENUMSET1:def 2;
then h.B in B by A76;
then mbc in INTERSECTION(B,C) by A83,SETFAM_1:def 5;
then mbc in INTERSECTION(B,C) \ {{}} by A92,XBOOLE_0:def 5;
then mbc in B '/\' C by PARTIT1:def 4;
then mbcd in INTERSECTION(B '/\' C,D) by A82,SETFAM_1:def 5;
then mbcd in INTERSECTION(B '/\' C,D) \ {{}} by A91,XBOOLE_0:def 5;
then
A93: mbcd in B '/\' C '/\' D by PARTIT1:def 4;
h.B /\ h.C /\ h.D /\ h.E c= xx
proof
let m be object;
assume
A94: m in h.B /\ h.C /\ h.D /\ h.E;
then
A95: m in h.B /\ h.C /\ h.D by XBOOLE_0:def 4;
then
A96: m in h.B /\ h.C by XBOOLE_0:def 4;
A97: rng h c= {h.B,h.C,h.D,h.E}
proof
let u be object;
assume u in rng h;
then consider x1 being object such that
A98: x1 in dom h and
A99: u = h.x1 by FUNCT_1:def 3;
now
per cases by A15,A17,A74,A98,ENUMSET1:def 2;
case
x1=B;
hence thesis by A99,ENUMSET1:def 2;
end;
case
x1=C;
hence thesis by A99,ENUMSET1:def 2;
end;
case
x1=D;
hence thesis by A99,ENUMSET1:def 2;
end;
case
x1=E;
hence thesis by A99,ENUMSET1:def 2;
end;
end;
hence thesis;
end;
for y being set holds y in rng h implies m in y
proof
let y be set;
assume
A100: y in rng h;
now
per cases by A97,A100,ENUMSET1:def 2;
case
y=h.B;
hence thesis by A96,XBOOLE_0:def 4;
end;
case
y=h.C;
hence thesis by A96,XBOOLE_0:def 4;
end;
case
y=h.D;
hence thesis by A95,XBOOLE_0:def 4;
end;
case
y=h.E;
hence thesis by A94,XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then m in meet (rng h) by A85,SETFAM_1:def 1;
hence thesis by A75,A77,A85,SETFAM_1:def 9;
end;
then ((h.B /\ h.C) /\ h.D) /\ h.E = x by A87,XBOOLE_0:def 10;
then x in INTERSECTION(B '/\' C '/\' D,E) by A81,A93,SETFAM_1:def 5;
then x in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A80,XBOOLE_0:def 5;
hence thesis by PARTIT1:def 4;
end;
then '/\' (G \ {A}) = B '/\' C '/\' D '/\' E by A19,XBOOLE_0:def 10;
hence thesis by BVFUNC_2:def 7;
end;
end;
theorem Th22:
G={A,B,C,D,E} & A<>B & B<>C & B<>D & B<>E implies CompF(B,G) = A
'/\' C '/\' D '/\' E
proof
assume that
A1: G={A,B,C,D,E} and
A2: A<>B & B<>C & B<>D & B<>E;
{A,B,C,D,E}={A,B} \/ {C,D,E} by ENUMSET1:8;
then G={B,A,C,D,E} by A1,ENUMSET1:8;
hence thesis by A2,Th21;
end;
theorem Th23:
G={A,B,C,D,E} & A<>C & B<>C & C<>D & C<>E implies CompF(C,G) = A
'/\' B '/\' D '/\' E
proof
assume that
A1: G={A,B,C,D,E} and
A2: A<>C & B<>C & C<>D & C<>E;
{A,B,C,D,E}={A,B,C} \/ {D,E} by ENUMSET1:9;
then {A,B,C,D,E}={A} \/ {B,C} \/ {D,E} by ENUMSET1:2;
then {A,B,C,D,E}={A,C,B} \/ {D,E} by ENUMSET1:2;
then {A,B,C,D,E}={A,C} \/ {B} \/ {D,E} by ENUMSET1:3;
then {A,B,C,D,E}={C,A,B} \/ {D,E} by ENUMSET1:3;
then G={C,A,B,D,E} by A1,ENUMSET1:9;
hence thesis by A2,Th21;
end;
theorem Th24:
G={A,B,C,D,E} & A<>D & B<>D & C<>D & D<>E implies CompF(D,G) = A
'/\' B '/\' C '/\' E
proof
assume that
A1: G={A,B,C,D,E} and
A2: A<>D & B<>D & C<>D & D<>E;
{A,B,C,D,E}={A,B} \/ {C,D,E} by ENUMSET1:8;
then {A,B,C,D,E}={A,B} \/ ({C,D} \/ {E}) by ENUMSET1:3;
then {A,B,C,D,E}={A,B} \/ {D,C,E} by ENUMSET1:3;
then G={A,B,D,C,E} by A1,ENUMSET1:8;
hence thesis by A2,Th23;
end;
theorem
G={A,B,C,D,E} & A<>E & B<>E & C<>E & D<>E implies CompF(E,G) = A '/\'
B '/\' C '/\' D
proof
assume that
A1: G={A,B,C,D,E} and
A2: A<>E & B<>E & C<>E & D<>E;
{A,B,C,D,E}={A,B,C} \/ {D,E} by ENUMSET1:9;
then G={A,B,C,E,D} by A1,ENUMSET1:9;
hence thesis by A2,Th24;
end;
theorem Th26:
for A,B,C,D,E being set, h being Function, A9,B9,C9,D9,E9 being
set st A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E & h
= (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (A .--> A9) holds
h.A = A9 & h.B = B9 & h.C = C9 & h.D = D9 & h.E = E9
proof
let A,B,C,D,E be set;
let h be Function;
let A9,B9,C9,D9,E9 be set;
assume that
A1: A<>B and
A2: A<>C and
A3: A<>D and
A4: A<>E and
A5: B<>C and
A6: B<>D and
A7: B<>E and
A8: C<>D and
A9: C<>E and
A10: D<>E and
A11: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (A
.--> A9);
A in dom (A .--> A9) by TARSKI:def 1;
then
A13: h.A = (A .--> A9).A by A11,FUNCT_4:13;
not C in dom (A .--> A9) by A2,TARSKI:def 1;
then
A14: h.C=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9)).C by A11,
FUNCT_4:11;
not B in dom (D .--> D9) by A6,TARSKI:def 1;
then
A16: ((B .--> B9) +* (C .--> C9) +* (D .--> D9)).B= ((B .--> B9) +* (C .-->
C9)).B by FUNCT_4:11;
not E in dom (A .--> A9) by A4,TARSKI:def 1;
then
A17: h.E=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9)).E by A11,
FUNCT_4:11;
E in dom (E .--> E9) by TARSKI:def 1;
then
A19: h.E=(E .--> E9).E by A17,FUNCT_4:13;
not C in dom (D .--> D9) by A8,TARSKI:def 1;
then
A20: ((B .--> B9) +* (C .--> C9) +* (D .--> D9)).C= ((B .--> B9) +* (C .-->
C9)).C by FUNCT_4:11;
not C in dom (E .--> E9) by A9,TARSKI:def 1;
then
A21: h.C=((B .--> B9) +* (C .--> C9) +* (D .--> D9)).C by A14,FUNCT_4:11;
C in dom (C .--> C9) by TARSKI:def 1;
then
A23: h.C=(C .--> C9).C by A21,A20,FUNCT_4:13;
not D in dom (A .--> A9) by A3,TARSKI:def 1;
then
A24: h.D=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9)).D by A11,
FUNCT_4:11;
not D in dom (E .--> E9) by A10,TARSKI:def 1;
then
A25: h.D=((B .--> B9) +* (C .--> C9) +* (D .--> D9)).D by A24,FUNCT_4:11;
D in dom (D .--> D9) by TARSKI:def 1;
then
A26: h.D=(D .--> D9).D by A25,FUNCT_4:13;
not B in dom (A .--> A9) by A1,TARSKI:def 1;
then
A27: h.B=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9)).B by A11,
FUNCT_4:11;
not B in dom (E .--> E9) by A7,TARSKI:def 1;
then
A28: h.B=((B .--> B9) +* (C .--> C9) +* (D .--> D9)).B by A27,FUNCT_4:11;
not B in dom (C .--> C9) by A5,TARSKI:def 1;
then h.B=(B .--> B9).B by A28,A16,FUNCT_4:11;
hence thesis by A13,A23,A26,A19,FUNCOP_1:72;
end;
theorem Th27:
for A,B,C,D,E being set, h being Function, A9,B9,C9,D9,E9 being
set st h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (A .-->
A9) holds dom h = {A,B,C,D,E}
proof
let A,B,C,D,E be set;
let h be Function;
let A9,B9,C9,D9,E9 be set;
assume
A1: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (A
.--> A9);
dom ((B .--> B9) +* (C .--> C9)) = dom (B .--> B9) \/ dom (C .--> C9) by
FUNCT_4:def 1;
then
dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9)) = dom (B .--> B9) \/ dom
(C .--> C9) \/ dom (D .--> D9) by FUNCT_4:def 1;
then
dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9)) = dom (B
.--> B9) \/ dom (C .--> C9) \/ dom (D .--> D9) \/ dom (E .--> E9) by
FUNCT_4:def 1;
then
A3: dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (A .-->
A9)) = dom (B .--> B9) \/ dom (C .--> C9) \/ dom (D .--> D9) \/ dom (E .--> E9)
\/ dom (A .--> A9) by FUNCT_4:def 1;
dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (A
.--> A9)) = {A} \/ (({B} \/ {C}) \/ {D} \/ {E}) by A3
.= {A} \/ ({B,C} \/ {D} \/ {E}) by ENUMSET1:1
.= {A} \/ ({B,C,D} \/ {E}) by ENUMSET1:3
.= {A} \/ {B,C,D,E} by ENUMSET1:6
.= {A,B,C,D,E} by ENUMSET1:7;
hence thesis by A1;
end;
theorem Th28:
for A,B,C,D,E being set, h being Function, A9,B9,C9,D9,E9 being
set st h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (A .-->
A9) holds rng h = {h.A,h.B,h.C,h.D,h.E}
proof
let A,B,C,D,E be set;
let h be Function;
let A9,B9,C9,D9,E9 be set;
assume
h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (A .--> A9 );
then
A1: dom h = {A,B,C,D,E} by Th27;
then
A2: B in dom h by ENUMSET1:def 3;
A3: D in dom h by A1,ENUMSET1:def 3;
A4: C in dom h by A1,ENUMSET1:def 3;
A5: rng h c= {h.A,h.B,h.C,h.D,h.E}
proof
let t be object;
assume t in rng h;
then consider x1 being object such that
A6: x1 in dom h and
A7: t = h.x1 by FUNCT_1:def 3;
now
per cases by A1,A6,ENUMSET1:def 3;
case
x1=A;
hence thesis by A7,ENUMSET1:def 3;
end;
case
x1=B;
hence thesis by A7,ENUMSET1:def 3;
end;
case
x1=C;
hence thesis by A7,ENUMSET1:def 3;
end;
case
x1=D;
hence thesis by A7,ENUMSET1:def 3;
end;
case
x1=E;
hence thesis by A7,ENUMSET1:def 3;
end;
end;
hence thesis;
end;
A8: E in dom h by A1,ENUMSET1:def 3;
A9: A in dom h by A1,ENUMSET1:def 3;
{h.A,h.B,h.C,h.D,h.E} c= rng h
proof
let t be object;
assume
A10: t in {h.A,h.B,h.C,h.D,h.E};
now
per cases by A10,ENUMSET1:def 3;
case
t=h.A;
hence thesis by A9,FUNCT_1:def 3;
end;
case
t=h.B;
hence thesis by A2,FUNCT_1:def 3;
end;
case
t=h.C;
hence thesis by A4,FUNCT_1:def 3;
end;
case
t=h.D;
hence thesis by A3,FUNCT_1:def 3;
end;
case
t=h.E;
hence thesis by A8,FUNCT_1:def 3;
end;
end;
hence thesis;
end;
hence thesis by A5,XBOOLE_0:def 10;
end;
theorem
for G being Subset of PARTITIONS(Y), A,B,C,D,E being a_partition of Y,
z,u being Element of Y, h being Function st G is independent & G={A,B,C,D,E} &
A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E holds
EqClass(u,B '/\' C '/\' D '/\' E) meets EqClass(z,A)
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E be a_partition of Y;
let z,u be Element of Y;
let h be Function;
assume that
A1: G is independent and
A2: G={A,B,C,D,E} and
A3: A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E;
set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (D .--> EqClass(u,
D)) +* (E .--> EqClass(u,E)) +* (A .--> EqClass(z,A));
A4: h.B = EqClass(u,B) by A3,Th26;
A5: h.D = EqClass(u,D) by A3,Th26;
A6: h.C = EqClass(u,C) by A3,Th26;
A7: h.E = EqClass(u,E) by A3,Th26;
A8: rng h = {h.A,h.B,h.C,h.D,h.E} by Th28;
rng h c= bool Y
proof
let t be object;
assume
A9: t in rng h;
now
per cases by A8,A9,ENUMSET1:def 3;
case
t=h.A;
then t=EqClass(z,A) by A3,Th26;
hence thesis;
end;
case
t=h.B;
hence thesis by A4;
end;
case
t=h.C;
hence thesis by A6;
end;
case
t=h.D;
hence thesis by A5;
end;
case
t=h.E;
hence thesis by A7;
end;
end;
hence thesis;
end;
then reconsider FF=rng h as Subset-Family of Y;
A10: dom h = G by A2,Th27;
for d being set st d in G holds h.d in d
proof
let d be set;
assume
A11: d in G;
now
per cases by A2,A11,ENUMSET1:def 3;
case
A12: d=A;
h.A=EqClass(z,A) by A3,Th26;
hence thesis by A12;
end;
case
A13: d=B;
h.B=EqClass(u,B) by A3,Th26;
hence thesis by A13;
end;
case
A14: d=C;
h.C=EqClass(u,C) by A3,Th26;
hence thesis by A14;
end;
case
A15: d=D;
h.D=EqClass(u,D) by A3,Th26;
hence thesis by A15;
end;
case
A16: d=E;
h.E=EqClass(u,E) by A3,Th26;
hence thesis by A16;
end;
end;
hence thesis;
end;
then (Intersect FF)<>{} by A1,A10,BVFUNC_2:def 5;
then consider m being object such that
A17: m in Intersect FF by XBOOLE_0:def 1;
A in dom h by A2,A10,ENUMSET1:def 3;
then
A18: h.A in rng h by FUNCT_1:def 3;
then
A19: m in meet FF by A17,SETFAM_1:def 9;
then
A20: m in h.A by A18,SETFAM_1:def 1;
D in dom h by A2,A10,ENUMSET1:def 3;
then h.D in rng h by FUNCT_1:def 3;
then
A21: m in h.D by A19,SETFAM_1:def 1;
C in dom h by A2,A10,ENUMSET1:def 3;
then h.C in rng h by FUNCT_1:def 3;
then
A22: m in h.C by A19,SETFAM_1:def 1;
B in dom h by A2,A10,ENUMSET1:def 3;
then h.B in rng h by FUNCT_1:def 3;
then m in h.B by A19,SETFAM_1:def 1;
then m in EqClass(u,B) /\ EqClass(u,C) by A4,A6,A22,XBOOLE_0:def 4;
then
A23: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A5,A21,XBOOLE_0:def 4
;
E in dom h by A2,A10,ENUMSET1:def 3;
then h.E in rng h by FUNCT_1:def 3;
then m in h.E by A19,SETFAM_1:def 1;
then
A24: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) by A7
,A23,XBOOLE_0:def 4;
set GG=EqClass(u,B '/\' C '/\' D '/\' E);
GG = EqClass(u,B '/\' C '/\' D) /\ EqClass(u,E) by Th1;
then
A25: GG = EqClass(u,B '/\' C) /\ EqClass(u,D) /\ EqClass(u,E) by Th1;
h.A = EqClass(z,A) by A3,Th26;
then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\
EqClass(z,A) by A20,A24,XBOOLE_0:def 4;
then EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) meets
EqClass(z,A) by XBOOLE_0:4;
hence thesis by A25,Th1;
end;
theorem
for G being Subset of PARTITIONS(Y), A,B,C,D,E being a_partition of Y,
z,u being Element of Y st G is independent & G={A,B,C,D,E} & A<>B & A<>C & A<>D
& A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E & EqClass(z,C '/\' D '/\' E)=
EqClass(u,C '/\' D '/\' E) holds EqClass(u,CompF(A,G)) meets EqClass(z,CompF(B,
G))
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E be a_partition of Y;
let z,u be Element of Y;
assume that
A1: G is independent and
A2: G={A,B,C,D,E} and
A3: A<>B and
A4: A<>C & A<>D & A<>E and
A5: B<>C & B<>D & B<>E and
A6: C<>D & C<>E & D<>E and
A7: EqClass(z,C '/\' D '/\' E)=EqClass(u,C '/\' D '/\' E);
set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (D .--> EqClass(u,
D)) +* (E .--> EqClass(u,E)) +* (A .--> EqClass(z,A));
A8: h.B = EqClass(u,B) by A3,A4,A5,A6,Th26;
A9: h.E = EqClass(u,E) by A3,A4,A5,A6,Th26;
A10: h.D = EqClass(u,D) by A3,A4,A5,A6,Th26;
A11: h.C = EqClass(u,C) by A3,A4,A5,A6,Th26;
A12: rng h = {h.A,h.B,h.C,h.D,h.E} by Th28;
rng h c= bool Y
proof
let t be object;
assume
A13: t in rng h;
now
per cases by A12,A13,ENUMSET1:def 3;
case
t=h.A;
then t=EqClass(z,A) by A3,A4,A5,A6,Th26;
hence thesis;
end;
case
t=h.B;
hence thesis by A8;
end;
case
t=h.C;
hence thesis by A11;
end;
case
t=h.D;
hence thesis by A10;
end;
case
t=h.E;
hence thesis by A9;
end;
end;
hence thesis;
end;
then reconsider FF=rng h as Subset-Family of Y;
A14: dom h = G by A2,Th27;
for d being set st d in G holds h.d in d
proof
let d be set;
assume
A15: d in G;
now
per cases by A2,A15,ENUMSET1:def 3;
case
A16: d=A;
h.A=EqClass(z,A) by A3,A4,A5,A6,Th26;
hence thesis by A16;
end;
case
A17: d=B;
h.B=EqClass(u,B) by A3,A4,A5,A6,Th26;
hence thesis by A17;
end;
case
A18: d=C;
h.C=EqClass(u,C) by A3,A4,A5,A6,Th26;
hence thesis by A18;
end;
case
A19: d=D;
h.D=EqClass(u,D) by A3,A4,A5,A6,Th26;
hence thesis by A19;
end;
case
A20: d=E;
h.E=EqClass(u,E) by A3,A4,A5,A6,Th26;
hence thesis by A20;
end;
end;
hence thesis;
end;
then (Intersect FF)<>{} by A1,A14,BVFUNC_2:def 5;
then consider m being object such that
A21: m in Intersect FF by XBOOLE_0:def 1;
A in dom h by A2,A14,ENUMSET1:def 3;
then
A22: h.A in rng h by FUNCT_1:def 3;
then
A23: m in meet FF by A21,SETFAM_1:def 9;
then
A24: m in h.A by A22,SETFAM_1:def 1;
D in dom h by A2,A14,ENUMSET1:def 3;
then h.D in rng h by FUNCT_1:def 3;
then
A25: m in h.D by A23,SETFAM_1:def 1;
C in dom h by A2,A14,ENUMSET1:def 3;
then h.C in rng h by FUNCT_1:def 3;
then
A26: m in h.C by A23,SETFAM_1:def 1;
B in dom h by A2,A14,ENUMSET1:def 3;
then h.B in rng h by FUNCT_1:def 3;
then m in h.B by A23,SETFAM_1:def 1;
then m in EqClass(u,B) /\ EqClass(u,C) by A8,A11,A26,XBOOLE_0:def 4;
then
A27: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A10,A25,
XBOOLE_0:def 4;
E in dom h by A2,A14,ENUMSET1:def 3;
then h.E in rng h by FUNCT_1:def 3;
then m in h.E by A23,SETFAM_1:def 1;
then
A28: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) by A9
,A27,XBOOLE_0:def 4;
set GG=EqClass(u,(((B '/\' C) '/\' D) '/\' E));
set I=EqClass(z,A);
GG = EqClass(u,B '/\' C '/\' D) /\ EqClass(u,E) by Th1;
then
A29: GG = EqClass(u,B '/\' C) /\ EqClass(u,D) /\ EqClass(u,E) by Th1;
h.A = EqClass(z,A) by A3,A4,A5,A6,Th26;
then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\
EqClass(z,A) by A24,A28,XBOOLE_0:def 4;
then GG /\ I <> {} by A29,Th1;
then consider p being object such that
A30: p in GG /\ I by XBOOLE_0:def 1;
reconsider p as Element of Y by A30;
set K=EqClass(p,C '/\' D '/\' E);
A31: p in GG by A30,XBOOLE_0:def 4;
A32: z in I by EQREL_1:def 6;
set L=EqClass(z,C '/\' D '/\' E);
A33: p in EqClass(p,C '/\' D '/\' E) by EQREL_1:def 6;
GG = EqClass(u,((B '/\' (C '/\' D)) '/\' E)) by PARTIT1:14;
then GG = EqClass(u,B '/\' (C '/\' D '/\' E)) by PARTIT1:14;
then GG c= L by A7,BVFUNC11:3;
then K meets L by A31,A33,XBOOLE_0:3;
then K=L by EQREL_1:41;
then z in K by EQREL_1:def 6;
then
A34: z in I /\ K by A32,XBOOLE_0:def 4;
set H=EqClass(z,CompF(B,G));
A '/\' (C '/\' D '/\' E) = A '/\' (C '/\' D) '/\' E by PARTIT1:14;
then
A35: A '/\' (C '/\' D '/\' E) = A '/\' C '/\' D '/\' E by PARTIT1:14;
A36: p in K & p in I by A30,EQREL_1:def 6,XBOOLE_0:def 4;
then p in I /\ K by XBOOLE_0:def 4;
then I /\ K in INTERSECTION(A,C '/\' D '/\' E) & not I /\ K in {{}} by
SETFAM_1:def 5,TARSKI:def 1;
then
A37: I /\ K in INTERSECTION(A,C '/\' D '/\' E) \ {{}} by XBOOLE_0:def 5;
CompF(B,G) = A '/\' C '/\' D '/\' E by A2,A3,A5,Th22;
then I /\ K in CompF(B,G) by A37,A35,PARTIT1:def 4;
then
A38: I /\ K = H or I /\ K misses H by EQREL_1:def 4;
z in H by EQREL_1:def 6;
then p in H by A36,A34,A38,XBOOLE_0:3,def 4;
then p in GG /\ H by A31,XBOOLE_0:def 4;
then GG meets H by XBOOLE_0:4;
hence thesis by A2,A3,A4,Th21;
end;
:: moved from BVFUNC23, AG 4.01.2006
theorem Th31:
G={A,B,C,D,E,F} & A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D
& B<>E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F implies CompF(A,G) = B
'/\' C '/\' D '/\' E '/\' F
proof
assume that
A1: G={A,B,C,D,E,F} and
A2: A<>B and
A3: A<>C and
A4: A<>D & A<>E and
A5: A<>F and
A6: B<>C & B<>D & B<>E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F;
A7: G \ {A}={A} \/ {B,C,D,E,F} \ {A} by A1,ENUMSET1:11
.= ({A} \ {A}) \/ ({B,C,D,E,F} \ {A}) by XBOOLE_1:42;
A8: not F in {A} by A5,TARSKI:def 1;
A9: ( not D in {A})& not E in {A} by A4,TARSKI:def 1;
A10: not C in {A} by A3,TARSKI:def 1;
A11: not B in {A} by A2,TARSKI:def 1;
A in {A} by TARSKI:def 1;
then
A12: {A} \ {A}={} by ZFMISC_1:60;
A13: {B,C,D,E,F} \ {A} = ({B} \/ {C,D,E,F}) \ {A} by ENUMSET1:7
.= ({B} \ {A}) \/ ({C,D,E,F} \ {A}) by XBOOLE_1:42
.= {B} \/ ({C,D,E,F} \ {A}) by A11,ZFMISC_1:59
.= {B} \/ (({C} \/ {D,E,F}) \ {A}) by ENUMSET1:4
.= {B} \/ (({C} \ {A}) \/ ({D,E,F} \ {A})) by XBOOLE_1:42
.= {B} \/ (({C} \ {A}) \/ (({D,E} \/ {F}) \ {A})) by ENUMSET1:3
.= {B} \/ (({C} \ {A}) \/ (({D,E} \ {A}) \/ ({F} \ {A}))) by XBOOLE_1:42
.= {B} \/ (({C} \ {A}) \/ ({D,E} \/ ({F} \ {A}))) by A9,ZFMISC_1:63
.= {B} \/ (({C} \ {A}) \/ ({D,E} \/ {F})) by A8,ZFMISC_1:59
.= {B} \/ ({C} \/ ({D,E} \/ {F})) by A10,ZFMISC_1:59
.= {B} \/ ({C} \/ {D,E,F}) by ENUMSET1:3
.= {B} \/ {C,D,E,F} by ENUMSET1:4
.= {B,C,D,E,F} by ENUMSET1:7;
A14: B '/\' C '/\' D '/\' E '/\' F c= '/\' (G \ {A})
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume
A15: x in B '/\' C '/\' D '/\' E '/\' F;
then
A16: x<>{} by EQREL_1:def 4;
x in INTERSECTION(B '/\' C '/\' D '/\' E,F) \ {{}} by A15,PARTIT1:def 4;
then consider bcde,f being set such that
A17: bcde in B '/\' C '/\' D '/\' E and
A18: f in F and
A19: x = bcde /\ f by SETFAM_1:def 5;
bcde in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A17,PARTIT1:def 4;
then consider bcd,e being set such that
A20: bcd in B '/\' C '/\' D and
A21: e in E and
A22: bcde = bcd /\ e by SETFAM_1:def 5;
bcd in INTERSECTION(B '/\' C,D) \ {{}} by A20,PARTIT1:def 4;
then consider bc,d being set such that
A23: bc in B '/\' C and
A24: d in D and
A25: bcd = bc /\ d by SETFAM_1:def 5;
bc in INTERSECTION(B,C) \ {{}} by A23,PARTIT1:def 4;
then consider b,c being set such that
A26: b in B and
A27: c in C and
A28: bc = b /\ c by SETFAM_1:def 5;
set h = (B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e) +* (F .--> f);
A29: h.B = b by A6,Th26;
A30: h.E = e by A6,Th26;
A31: h.F = f by A6,Th26;
A32: dom ((B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e) +* (F .-->
f)) = {F,B,C,D,E} by Th27
.= {F} \/ {B,C,D,E} by ENUMSET1:7
.= {B,C,D,E,F} by ENUMSET1:10;
then
A33: C in dom h by ENUMSET1:def 3;
A34: F in dom h by A32,ENUMSET1:def 3;
A35: E in dom h by A32,ENUMSET1:def 3;
A36: h.C = c by A6,Th26;
A37: rng h c= {h.D,h.B,h.C,h.E,h.F}
proof
let t be object;
assume t in rng h;
then consider x1 being object such that
A38: x1 in dom h and
A39: t = h.x1 by FUNCT_1:def 3;
now
per cases by A32,A38,ENUMSET1:def 3;
case
x1=D;
hence thesis by A39,ENUMSET1:def 3;
end;
case
x1=B;
hence thesis by A39,ENUMSET1:def 3;
end;
case
x1=C;
hence thesis by A39,ENUMSET1:def 3;
end;
case
x1=E;
hence thesis by A39,ENUMSET1:def 3;
end;
case
x1=F;
hence thesis by A39,ENUMSET1:def 3;
end;
end;
hence thesis;
end;
A40: h.D = d by A6,Th26;
rng h c= bool Y
proof
let t be object;
assume
A41: t in rng h;
now
per cases by A37,A41,ENUMSET1:def 3;
case
t=h.D;
hence thesis by A24,A40;
end;
case
t=h.B;
hence thesis by A26,A29;
end;
case
t=h.C;
hence thesis by A27,A36;
end;
case
t=h.E;
hence thesis by A21,A30;
end;
case
t=h.F;
hence thesis by A18,A31;
end;
end;
hence thesis;
end;
then reconsider FF=rng h as Subset-Family of Y;
A42: D in dom h by A32,ENUMSET1:def 3;
then h.D in rng h by FUNCT_1:def 3;
then
A43: Intersect FF = meet (rng h) by SETFAM_1:def 9;
A44: B in dom h by A32,ENUMSET1:def 3;
{h.D,h.B,h.C,h.E,h.F} c= rng h
proof
let t be object;
assume
A45: t in {h.D,h.B,h.C,h.E,h.F};
now
per cases by A45,ENUMSET1:def 3;
case
t=h.D;
hence thesis by A42,FUNCT_1:def 3;
end;
case
t=h.B;
hence thesis by A44,FUNCT_1:def 3;
end;
case
t=h.C;
hence thesis by A33,FUNCT_1:def 3;
end;
case
t=h.E;
hence thesis by A35,FUNCT_1:def 3;
end;
case
t=h.F;
hence thesis by A34,FUNCT_1:def 3;
end;
end;
hence thesis;
end;
then
A46: rng h = {h.D,h.B,h.C,h.E,h.F} by A37,XBOOLE_0:def 10;
A47: xx c= Intersect FF
proof
let u be object;
assume
A48: u in xx;
for y be set holds y in FF implies u in y
proof
let y be set;
assume
A49: y in FF;
now
per cases by A37,A49,ENUMSET1:def 3;
case
A50: y=h.D;
u in (d /\ ((b /\ c) /\ e)) /\ f by A19,A22,A25,A28,A48,XBOOLE_1:16
;
then
A51: u in d /\ ((b /\ c) /\ e /\ f) by XBOOLE_1:16;
y=d by A6,A50,Th26;
hence thesis by A51,XBOOLE_0:def 4;
end;
case
A52: y=h.B;
u in (c /\ (d /\ b)) /\ e /\ f by A19,A22,A25,A28,A48,XBOOLE_1:16;
then u in c /\ ((d /\ b) /\ e) /\ f by XBOOLE_1:16;
then u in c /\ ((d /\ e) /\ b) /\ f by XBOOLE_1:16;
then u in c /\ (((d /\ e) /\ b) /\ f) by XBOOLE_1:16;
then u in c /\ ((d /\ e) /\ (f /\ b)) by XBOOLE_1:16;
then u in (c /\ (d /\ e)) /\ (f /\ b) by XBOOLE_1:16;
then
A53: u in ((c /\ (d /\ e)) /\ f) /\ b by XBOOLE_1:16;
y=b by A6,A52,Th26;
hence thesis by A53,XBOOLE_0:def 4;
end;
case
A54: y=h.C;
u in ((c /\ (b /\ d)) /\ e) /\ f by A19,A22,A25,A28,A48,XBOOLE_1:16
;
then u in (c /\ ((b /\ d) /\ e)) /\ f by XBOOLE_1:16;
then
A55: u in c /\ (((b /\ d) /\ e) /\ f) by XBOOLE_1:16;
y=c by A6,A54,Th26;
hence thesis by A55,XBOOLE_0:def 4;
end;
case
y=h.E;
then
A56: y=e by A6,Th26;
u in (((b /\ c) /\ d) /\ f) /\ e by A19,A22,A25,A28,A48,XBOOLE_1:16
;
hence thesis by A56,XBOOLE_0:def 4;
end;
case
y=h.F;
hence thesis by A19,A31,A48,XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u in meet FF by A46,SETFAM_1:def 1;
hence thesis by A46,SETFAM_1:def 9;
end;
A57: for p being set st p in (G \ {A}) holds h.p in p
proof
let p be set;
assume
A58: p in (G \ {A});
now
per cases by A7,A12,A58,ENUMSET1:def 3;
case
p=D;
hence thesis by A6,A24,Th26;
end;
case
p=B;
hence thesis by A6,A26,Th26;
end;
case
p=C;
hence thesis by A6,A27,Th26;
end;
case
p=E;
hence thesis by A6,A21,Th26;
end;
case
p=F;
hence thesis by A6,A18,Th26;
end;
end;
hence thesis;
end;
Intersect FF c= xx
proof
let t be object;
assume
A59: t in Intersect FF;
h.C in rng h by A46,ENUMSET1:def 3;
then
A60: t in c by A36,A43,A59,SETFAM_1:def 1;
h.B in rng h by A46,ENUMSET1:def 3;
then t in b by A29,A43,A59,SETFAM_1:def 1;
then
A61: t in b /\ c by A60,XBOOLE_0:def 4;
h.D in rng h by A46,ENUMSET1:def 3;
then t in d by A40,A43,A59,SETFAM_1:def 1;
then
A62: t in (b /\ c) /\ d by A61,XBOOLE_0:def 4;
h.E in rng h by A46,ENUMSET1:def 3;
then t in e by A30,A43,A59,SETFAM_1:def 1;
then
A63: t in (b /\ c) /\ d /\ e by A62,XBOOLE_0:def 4;
h.F in rng h by A46,ENUMSET1:def 3;
then t in f by A31,A43,A59,SETFAM_1:def 1;
hence thesis by A19,A22,A25,A28,A63,XBOOLE_0:def 4;
end;
then x = Intersect FF by A47,XBOOLE_0:def 10;
hence thesis by A7,A13,A12,A32,A57,A16,BVFUNC_2:def 1;
end;
A64: '/\' (G \ {A}) c= B '/\' C '/\' D '/\' E '/\' F
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume x in '/\' (G \ {A});
then consider h being Function, FF being Subset-Family of Y such that
A65: dom h=(G \ {A}) and
A66: rng h = FF and
A67: for d being set st d in (G \ {A}) holds h.d in d and
A68: x=Intersect FF and
A69: x<>{} by BVFUNC_2:def 1;
A70: C in (G \ {A}) by A7,A13,A12,ENUMSET1:def 3;
then
A71: h.C in C by A67;
set mbc=h.B /\ h.C;
A72: B in (G \ {A}) by A7,A13,A12,ENUMSET1:def 3;
then h.B in B by A67;
then
A73: mbc in INTERSECTION(B,C) by A71,SETFAM_1:def 5;
set mbcd=(h.B /\ h.C) /\ h.D;
A74: E in (G \ {A}) by A7,A13,A12,ENUMSET1:def 3;
then
A75: h.E in rng h by A65,FUNCT_1:def 3;
A76: h.B in rng h by A65,A72,FUNCT_1:def 3;
then
A77: Intersect FF = meet (rng h) by A66,SETFAM_1:def 9;
A78: h.C in rng h by A65,A70,FUNCT_1:def 3;
A79: F in (G \ {A}) by A7,A13,A12,ENUMSET1:def 3;
then
A80: h.F in rng h by A65,FUNCT_1:def 3;
A81: D in (G \ {A}) by A7,A13,A12,ENUMSET1:def 3;
then
A82: h.D in rng h by A65,FUNCT_1:def 3;
A83: xx c= h.B /\ h.C /\ h.D /\ h.E /\ h.F
proof
let m be object;
assume
A84: m in xx;
then m in h.B & m in h.C by A68,A76,A78,A77,SETFAM_1:def 1;
then
A85: m in h.B /\ h.C by XBOOLE_0:def 4;
m in h.D by A68,A82,A77,A84,SETFAM_1:def 1;
then
A86: m in h.B /\ h.C /\ h.D by A85,XBOOLE_0:def 4;
m in h.E by A68,A75,A77,A84,SETFAM_1:def 1;
then
A87: m in h.B /\ h.C /\ h.D /\ h.E by A86,XBOOLE_0:def 4;
m in h.F by A68,A80,A77,A84,SETFAM_1:def 1;
hence thesis by A87,XBOOLE_0:def 4;
end;
then mbcd<>{} by A69;
then
A88: not mbcd in {{}} by TARSKI:def 1;
A89: rng h <> {} by A65,A72,FUNCT_1:3;
h.B /\ h.C /\ h.D /\ h.E /\ h.F c= xx
proof
let m be object;
assume
A90: m in h.B /\ h.C /\ h.D /\ h.E /\ h.F;
then
A91: m in h.B /\ h.C /\ h.D /\ h.E by XBOOLE_0:def 4;
then
A92: m in h.B /\ h.C /\ h.D by XBOOLE_0:def 4;
then
A93: m in h.B /\ h.C by XBOOLE_0:def 4;
A94: rng h c= {h.B,h.C,h.D,h.E,h.F}
proof
let u be object;
assume u in rng h;
then consider x1 being object such that
A95: x1 in dom h and
A96: u = h.x1 by FUNCT_1:def 3;
now
per cases by A7,A12,A65,A95,ENUMSET1:def 3;
case
x1=B;
hence thesis by A96,ENUMSET1:def 3;
end;
case
x1=C;
hence thesis by A96,ENUMSET1:def 3;
end;
case
x1=D;
hence thesis by A96,ENUMSET1:def 3;
end;
case
x1=E;
hence thesis by A96,ENUMSET1:def 3;
end;
case
x1=F;
hence thesis by A96,ENUMSET1:def 3;
end;
end;
hence thesis;
end;
for y being set holds y in rng h implies m in y
proof
let y be set;
assume
A97: y in rng h;
now
per cases by A94,A97,ENUMSET1:def 3;
case
y=h.B;
hence thesis by A93,XBOOLE_0:def 4;
end;
case
y=h.C;
hence thesis by A93,XBOOLE_0:def 4;
end;
case
y=h.D;
hence thesis by A92,XBOOLE_0:def 4;
end;
case
y=h.E;
hence thesis by A91,XBOOLE_0:def 4;
end;
case
y=h.F;
hence thesis by A90,XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
hence thesis by A68,A89,A77,SETFAM_1:def 1;
end;
then
A98: ((h.B /\ h.C) /\ h.D) /\ h.E /\ h.F = x by A83,XBOOLE_0:def 10;
mbc<>{} by A69,A83;
then not mbc in {{}} by TARSKI:def 1;
then mbc in INTERSECTION(B,C) \ {{}} by A73,XBOOLE_0:def 5;
then
A99: mbc in B '/\' C by PARTIT1:def 4;
h.D in D by A67,A81;
then mbcd in INTERSECTION(B '/\' C,D) by A99,SETFAM_1:def 5;
then mbcd in INTERSECTION(B '/\' C,D) \ {{}} by A88,XBOOLE_0:def 5;
then
A100: mbcd in B '/\' C '/\' D by PARTIT1:def 4;
set mbcde=(h.B /\ h.C) /\ h.D /\ h.E;
A101: not x in {{}} by A69,TARSKI:def 1;
mbcde<>{} by A69,A83;
then
A102: not mbcde in {{}} by TARSKI:def 1;
h.E in E by A67,A74;
then mbcde in INTERSECTION(B '/\' C '/\' D,E) by A100,SETFAM_1:def 5;
then mbcde in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A102,XBOOLE_0:def 5
;
then
A103: mbcde in (B '/\' C '/\' D '/\' E) by PARTIT1:def 4;
h.F in F by A67,A79;
then x in INTERSECTION(B '/\' C '/\' D '/\' E,F) by A98,A103,SETFAM_1:def 5
;
then x in INTERSECTION(B '/\' C '/\' D '/\' E,F) \ {{}} by A101,
XBOOLE_0:def 5;
hence thesis by PARTIT1:def 4;
end;
CompF(A,G)='/\' (G \ {A}) by BVFUNC_2:def 7;
hence thesis by A14,A64,XBOOLE_0:def 10;
end;
theorem Th32:
G={A,B,C,D,E,F} & A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D
& B<>E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F implies CompF(B,G) = A
'/\' C '/\' D '/\' E '/\' F
proof
assume that
A1: G={A,B,C,D,E,F} and
A2: A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D & B<>E & B<>F & C<>D
& C<>E & C<>F & D<>E & D<>F & E<>F;
{A,B,C,D,E,F}={B,A} \/ {C,D,E,F} by ENUMSET1:12;
then G={B,A,C,D,E,F} by A1,ENUMSET1:12;
hence thesis by A2,Th31;
end;
theorem Th33:
G={A,B,C,D,E,F} & A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D
& B<>E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F implies CompF(C,G) = A
'/\' B '/\' D '/\' E '/\' F
proof
A1: {A,B,C,D,E,F}={A,B,C} \/ {D,E,F} by ENUMSET1:13
.={A} \/ {B,C} \/ {D,E,F} by ENUMSET1:2
.={A,C,B} \/ {D,E,F} by ENUMSET1:2
.={A,C,B,D,E,F} by ENUMSET1:13;
assume G={A,B,C,D,E,F} & A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D & B
<>E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F;
hence thesis by A1,Th32;
end;
theorem Th34:
G={A,B,C,D,E,F} & A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D
& B<>E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F implies CompF(D,G) = A
'/\' B '/\' C '/\' E '/\' F
proof
A1: {A,B,C,D,E,F} ={A,B} \/ {C,D,E,F} by ENUMSET1:12
.={A,B} \/ ({C,D} \/ {E,F}) by ENUMSET1:5
.={A,B} \/ {D,C,E,F} by ENUMSET1:5
.={A,B,D,C,E,F} by ENUMSET1:12;
assume G={A,B,C,D,E,F} & A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D & B
<>E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F;
hence thesis by A1,Th33;
end;
theorem Th35:
G={A,B,C,D,E,F} & A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D
& B<>E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F implies CompF(E,G) = A
'/\' B '/\' C '/\' D '/\' F
proof
A1: {A,B,C,D,E,F} ={A,B,C} \/ {D,E,F} by ENUMSET1:13
.={A,B,C} \/ ({D,E} \/ {F}) by ENUMSET1:3
.={A,B,C} \/ {E,D,F} by ENUMSET1:3
.={A,B,C,E,D,F} by ENUMSET1:13;
assume G={A,B,C,D,E,F} & A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D & B
<>E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F;
hence thesis by A1,Th34;
end;
theorem
G={A,B,C,D,E,F} & A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D & B<>
E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F implies CompF(F,G) = A '/\'
B '/\' C '/\' D '/\' E
proof
A1: {A,B,C,D,E,F} ={A,B,C,D} \/ {E,F} by ENUMSET1:14
.={A,B,C,D,F,E} by ENUMSET1:14;
assume G={A,B,C,D,E,F} & A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D & B
<>E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F;
hence thesis by A1,Th35;
end;
theorem Th37:
for A,B,C,D,E,F being set, h being Function, A9,B9,C9,D9,E9,F9
being set st A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D & B<>E & B<>F & C<>
D & C<>E & C<>F & D<>E & D<>F & E<>F & h = (B .--> B9) +* (C .--> C9) +* (D
.--> D9) +* (E .--> E9) +* (F .--> F9) +* (A .--> A9) holds h.A = A9 & h.B = B9
& h.C = C9 & h.D = D9 & h.E = E9 & h.F = F9
proof
let A,B,C,D,E,F be set;
let h be Function;
let A9,B9,C9,D9,E9,F9 be set;
assume that
A1: A<>B and
A2: A<>C and
A3: A<>D and
A4: A<>E and
A5: A<>F and
A6: B<>C & B<>D & B<>E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F and
A7: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (A .--> A9);
A in dom (A .--> A9) by TARSKI:def 1;
then
A9: h.A = (A .--> A9).A by A7,FUNCT_4:13;
not C in dom (A .--> A9) by A2,TARSKI:def 1;
then
A10: h.C=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9)).C by A7,FUNCT_4:11;
not F in dom (A .--> A9) by A5,TARSKI:def 1;
then
A11: h.F=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9)).F by A7,FUNCT_4:11
.= F9 by A6,Th26;
not E in dom (A .--> A9) by A4,TARSKI:def 1;
then
A12: h.E=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9)).E by A7,FUNCT_4:11
.= E9 by A6,Th26;
not D in dom (A .--> A9) by A3,TARSKI:def 1;
then
A13: h.D=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9)).D by A7,FUNCT_4:11
.= D9 by A6,Th26;
not B in dom (A .--> A9) by A1,TARSKI:def 1;
then h.B=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9)).B by A7,FUNCT_4:11
.= B9 by A6,Th26;
hence thesis by A6,A9,A10,A13,A12,A11,Th26,FUNCOP_1:72;
end;
theorem Th38:
for A,B,C,D,E,F being set, h being Function, A9,B9,C9,D9,E9,F9
being set st h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (A .--> A9) holds dom h = {A,B,C,D,E,F}
proof
let A,B,C,D,E,F be set;
let h be Function;
let A9,B9,C9,D9,E9,F9 be set;
assume
A1: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (A .--> A9);
A2: dom (A .--> A9) = {A};
dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F .-->
F9)) = {F,B,C,D,E} by Th27
.= {F} \/ {B,C,D,E} by ENUMSET1:7
.= {B,C,D,E,F} by ENUMSET1:10;
then
dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F .-->
F9) +*(A .--> A9)) = {B,C,D,E,F} \/ {A} by A2,FUNCT_4:def 1
.= {A,B,C,D,E,F} by ENUMSET1:11;
hence thesis by A1;
end;
theorem Th39:
for A,B,C,D,E,F being set, h being Function, A9,B9,C9,D9,E9,F9
being set st h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (A .--> A9) holds rng h = {h.A,h.B,h.C,h.D,h.E,h.F}
proof
let A,B,C,D,E,F be set;
let h be Function;
let A9,B9,C9,D9,E9,F9 be set;
assume h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (A .--> A9);
then
A1: dom h={A,B,C,D,E,F} by Th38;
then
A2: B in dom h by ENUMSET1:def 4;
A3: rng h c= {h.A,h.B,h.C,h.D,h.E,h.F}
proof
let t be object;
assume t in rng h;
then consider x1 being object such that
A4: x1 in dom h and
A5: t = h.x1 by FUNCT_1:def 3;
now
per cases by A1,A4,ENUMSET1:def 4;
case
x1=A;
hence thesis by A5,ENUMSET1:def 4;
end;
case
x1=B;
hence thesis by A5,ENUMSET1:def 4;
end;
case
x1=C;
hence thesis by A5,ENUMSET1:def 4;
end;
case
x1=D;
hence thesis by A5,ENUMSET1:def 4;
end;
case
x1=E;
hence thesis by A5,ENUMSET1:def 4;
end;
case
x1=F;
hence thesis by A5,ENUMSET1:def 4;
end;
end;
hence thesis;
end;
A6: D in dom h by A1,ENUMSET1:def 4;
A7: C in dom h by A1,ENUMSET1:def 4;
A8: F in dom h by A1,ENUMSET1:def 4;
A9: E in dom h by A1,ENUMSET1:def 4;
A10: A in dom h by A1,ENUMSET1:def 4;
{h.A,h.B,h.C,h.D,h.E,h.F} c= rng h
proof
let t be object;
assume
A11: t in {h.A,h.B,h.C,h.D,h.E,h.F};
now
per cases by A11,ENUMSET1:def 4;
case
t=h.A;
hence thesis by A10,FUNCT_1:def 3;
end;
case
t=h.B;
hence thesis by A2,FUNCT_1:def 3;
end;
case
t=h.C;
hence thesis by A7,FUNCT_1:def 3;
end;
case
t=h.D;
hence thesis by A6,FUNCT_1:def 3;
end;
case
t=h.E;
hence thesis by A9,FUNCT_1:def 3;
end;
case
t=h.F;
hence thesis by A8,FUNCT_1:def 3;
end;
end;
hence thesis;
end;
hence thesis by A3,XBOOLE_0:def 10;
end;
theorem
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F being a_partition of
Y, z,u being Element of Y, h being Function st G is independent & G={A,B,C,D,E,
F} & A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D & B<>E & B<>F & C<>D & C<>E
& C<>F & D<>E & D<>F & E<>F holds EqClass(u,B '/\' C '/\' D '/\' E '/\' F)
meets EqClass(z,A)
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F be a_partition of Y;
let z,u be Element of Y;
let h be Function;
assume that
A1: G is independent and
A2: G={A,B,C,D,E,F} and
A3: A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D & B<>E & B<>F & C<>D
& C<>E & C<>F & D<>E & D<>F & E<>F;
set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (D .--> EqClass(u,
D)) +* (E .--> EqClass(u,E)) +* (F .--> EqClass(u,F)) +* (A .--> EqClass(z,A));
A4: h.A = EqClass(z,A) by A3,Th37;
set GG=EqClass(u,B '/\' C '/\' D '/\' E '/\' F);
GG = EqClass(u,B '/\' C '/\' D '/\' E) /\ EqClass(u,F) by Th1;
then GG = EqClass(u,B '/\' C '/\' D) /\ EqClass(u,E) /\ EqClass(u,F) by Th1;
then
GG = EqClass(u,B '/\' C) /\ EqClass(u,D) /\ EqClass(u,E) /\ EqClass (u,
F) by Th1;
then
A5: GG /\ EqClass(z,A) = ((((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u, D)
) /\ EqClass(u,E)) /\ EqClass(u,F)) /\ EqClass(z,A) by Th1;
A6: h.B = EqClass(u,B) by A3,Th37;
A7: h.D = EqClass(u,D) by A3,Th37;
A8: h.C = EqClass(u,C) by A3,Th37;
A9: h.F = EqClass(u,F) by A3,Th37;
A10: h.E = EqClass(u,E) by A3,Th37;
A11: rng h = {h.A,h.B,h.C,h.D,h.E,h.F} by Th39;
rng h c= bool Y
proof
let t be object;
assume
A12: t in rng h;
now
per cases by A11,A12,ENUMSET1:def 4;
case
t=h.A;
hence thesis by A4;
end;
case
t=h.B;
hence thesis by A6;
end;
case
t=h.C;
hence thesis by A8;
end;
case
t=h.D;
hence thesis by A7;
end;
case
t=h.E;
hence thesis by A10;
end;
case
t=h.F;
hence thesis by A9;
end;
end;
hence thesis;
end;
then reconsider FF=rng h as Subset-Family of Y;
A13: dom h = G by A2,Th38;
then A in dom h by A2,ENUMSET1:def 4;
then
A14: h.A in rng h by FUNCT_1:def 3;
then
A15: Intersect FF = meet rng h by SETFAM_1:def 9;
for d being set st d in G holds h.d in d
proof
let d be set;
assume
A16: d in G;
now
per cases by A2,A16,ENUMSET1:def 4;
case
d=A;
hence thesis by A4;
end;
case
d=B;
hence thesis by A6;
end;
case
d=C;
hence thesis by A8;
end;
case
d=D;
hence thesis by A7;
end;
case
d=E;
hence thesis by A10;
end;
case
d=F;
hence thesis by A9;
end;
end;
hence thesis;
end;
then Intersect FF <> {} by A1,A13,BVFUNC_2:def 5;
then consider m being object such that
A17: m in Intersect FF by XBOOLE_0:def 1;
C in dom h by A2,A13,ENUMSET1:def 4;
then h.C in rng h by FUNCT_1:def 3;
then
A18: m in EqClass(u,C) by A8,A15,A17,SETFAM_1:def 1;
B in dom h by A2,A13,ENUMSET1:def 4;
then h.B in rng h by FUNCT_1:def 3;
then m in EqClass(u,B) by A6,A15,A17,SETFAM_1:def 1;
then
A19: m in EqClass(u,B) /\ EqClass(u,C) by A18,XBOOLE_0:def 4;
D in dom h by A2,A13,ENUMSET1:def 4;
then h.D in rng h by FUNCT_1:def 3;
then m in EqClass(u,D) by A7,A15,A17,SETFAM_1:def 1;
then
A20: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A19,XBOOLE_0:def 4;
E in dom h by A2,A13,ENUMSET1:def 4;
then h.E in rng h by FUNCT_1:def 3;
then m in EqClass(u,E) by A10,A15,A17,SETFAM_1:def 1;
then
A21: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) by A20,
XBOOLE_0:def 4;
F in dom h by A2,A13,ENUMSET1:def 4;
then h.F in rng h by FUNCT_1:def 3;
then m in EqClass(u,F) by A9,A15,A17,SETFAM_1:def 1;
then
A22: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\
EqClass(u,F) by A21,XBOOLE_0:def 4;
m in EqClass(z,A) by A4,A14,A15,A17,SETFAM_1:def 1;
then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\
EqClass(u,F) /\ EqClass(z,A) by A22,XBOOLE_0:def 4;
hence thesis by A5,XBOOLE_0:def 7;
end;
theorem
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F being a_partition of
Y, z,u being Element of Y, h being Function st G is independent & G={A,B,C,D,E,
F} & A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D & B<>E & B<>F & C<>D & C<>E
& C<>F & D<>E & D<>F & E<>F & EqClass(z,C '/\' D '/\' E '/\' F)=EqClass(u,C
'/\' D '/\' E '/\' F) holds EqClass(u,CompF(A,G)) meets EqClass(z,CompF(B,G))
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F be a_partition of Y;
let z,u be Element of Y;
let h be Function;
assume that
A1: G is independent and
A2: G={A,B,C,D,E,F} and
A3: A<>B & A<>C & A<>D & A<>E & A<>F & B<>C & B<>D & B<>E & B<>F & C<>D
& C<>E & C<>F & D<>E & D<>F & E<>F and
A4: EqClass(z,C '/\' D '/\' E '/\' F)=EqClass(u,C '/\' D '/\' E '/\' F);
set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (D .--> EqClass(u,
D)) +* (E .--> EqClass(u,E)) +* (F .--> EqClass(u,F)) +* (A .--> EqClass(z,A));
A5: h.A = EqClass(z,A) by A3,Th37;
set I=EqClass(z,A), GG=EqClass(u,(((B '/\' C) '/\' D) '/\' E '/\' F));
set H=EqClass(z,CompF(B,G));
A6: A '/\' (C '/\' D '/\' E '/\' F) = A '/\' (C '/\' D '/\' E) '/\' F by
PARTIT1:14
.= A '/\' (C '/\' D) '/\' E '/\' F by PARTIT1:14
.= A '/\' C '/\' D '/\' E '/\' F by PARTIT1:14;
GG = EqClass(u,B '/\' C '/\' D '/\' E) /\ EqClass(u,F) by Th1;
then GG = EqClass(u,B '/\' C '/\' D) /\ EqClass(u,E) /\ EqClass(u,F) by Th1;
then
GG = EqClass(u,B '/\' C) /\ EqClass(u,D) /\ EqClass(u,E) /\ EqClass (u,
F) by Th1;
then
A7: GG /\ I = ((((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass
(u,E)) /\ EqClass(u,F)) /\ EqClass(z,A) by Th1;
A8: h.B = EqClass(u,B) by A3,Th37;
A9: h.F = EqClass(u,F) by A3,Th37;
A10: h.E = EqClass(u,E) by A3,Th37;
A11: h.D = EqClass(u,D) by A3,Th37;
A12: h.C = EqClass(u,C) by A3,Th37;
A13: rng h = {h.A,h.B,h.C,h.D,h.E,h.F} by Th39;
rng h c= bool Y
proof
let t be object;
assume
A14: t in rng h;
now
per cases by A13,A14,ENUMSET1:def 4;
case
t=h.A;
hence thesis by A5;
end;
case
t=h.B;
hence thesis by A8;
end;
case
t=h.C;
hence thesis by A12;
end;
case
t=h.D;
hence thesis by A11;
end;
case
t=h.E;
hence thesis by A10;
end;
case
t=h.F;
hence thesis by A9;
end;
end;
hence thesis;
end;
then reconsider FF=rng h as Subset-Family of Y;
A15: dom h = G by A2,Th38;
then A in dom h by A2,ENUMSET1:def 4;
then
A16: h.A in rng h by FUNCT_1:def 3;
then
A17: Intersect FF = meet (rng h) by SETFAM_1:def 9;
for d being set st d in G holds h.d in d
proof
let d be set;
assume
A18: d in G;
now
per cases by A2,A18,ENUMSET1:def 4;
case
d=A;
hence thesis by A5;
end;
case
d=B;
hence thesis by A8;
end;
case
d=C;
hence thesis by A12;
end;
case
d=D;
hence thesis by A11;
end;
case
d=E;
hence thesis by A10;
end;
case
d=F;
hence thesis by A9;
end;
end;
hence thesis;
end;
then (Intersect FF)<>{} by A1,A15,BVFUNC_2:def 5;
then consider m being object such that
A19: m in Intersect FF by XBOOLE_0:def 1;
D in dom h by A2,A15,ENUMSET1:def 4;
then h.D in rng h by FUNCT_1:def 3;
then m in h.D by A17,A19,SETFAM_1:def 1;
then
A20: m in EqClass(u,D) by A3,Th37;
C in dom h by A2,A15,ENUMSET1:def 4;
then h.C in rng h by FUNCT_1:def 3;
then m in h.C by A17,A19,SETFAM_1:def 1;
then
A21: m in EqClass(u,C) by A3,Th37;
B in dom h by A2,A15,ENUMSET1:def 4;
then h.B in rng h by FUNCT_1:def 3;
then m in h.B by A17,A19,SETFAM_1:def 1;
then m in EqClass(u,B) by A3,Th37;
then m in EqClass(u,B) /\ EqClass(u,C) by A21,XBOOLE_0:def 4;
then
A22: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A20,XBOOLE_0:def 4;
F in dom h by A2,A15,ENUMSET1:def 4;
then h.F in rng h by FUNCT_1:def 3;
then m in h.F by A17,A19,SETFAM_1:def 1;
then
A23: m in EqClass(u,F) by A3,Th37;
E in dom h by A2,A15,ENUMSET1:def 4;
then h.E in rng h by FUNCT_1:def 3;
then m in h.E by A17,A19,SETFAM_1:def 1;
then m in EqClass(u,E) by A3,Th37;
then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) by A22
,XBOOLE_0:def 4;
then
A24: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\
EqClass(u,F) by A23,XBOOLE_0:def 4;
m in h.A by A16,A17,A19,SETFAM_1:def 1;
then m in EqClass(z,A) by A3,Th37;
then GG /\ I <> {} by A7,A24,XBOOLE_0:def 4;
then consider p being object such that
A25: p in GG /\ I by XBOOLE_0:def 1;
reconsider p as Element of Y by A25;
A26: p in GG by A25,XBOOLE_0:def 4;
set L=EqClass(z,C '/\' D '/\' E '/\' F);
GG = EqClass(u,(((B '/\' (C '/\' D)) '/\' E) '/\' F)) by PARTIT1:14;
then GG = EqClass(u,((B '/\' ((C '/\' D) '/\' E)) '/\' F)) by PARTIT1:14;
then GG = EqClass(u,B '/\' (C '/\' D '/\' E '/\' F)) by PARTIT1:14;
then
A27: GG c= L by A4,BVFUNC11:3;
A28: z in H by EQREL_1:def 6;
set K=EqClass(p,C '/\' D '/\' E '/\' F);
p in K & p in I by A25,EQREL_1:def 6,XBOOLE_0:def 4;
then
A29: p in I /\ K by XBOOLE_0:def 4;
then I /\ K in INTERSECTION(A,C '/\' D '/\' E '/\' F) & not I /\ K in {{}}
by SETFAM_1:def 5,TARSKI:def 1;
then I /\ K in INTERSECTION(A,C '/\' D '/\' E '/\' F) \ {{}} by
XBOOLE_0:def 5;
then
A30: I /\ K in A '/\' (C '/\' D '/\' E '/\' F) by PARTIT1:def 4;
p in EqClass(p,C '/\' D '/\' E '/\' F) by EQREL_1:def 6;
then K meets L by A27,A26,XBOOLE_0:3;
then K=L by EQREL_1:41;
then
A31: z in K by EQREL_1:def 6;
z in I by EQREL_1:def 6;
then
A32: z in I /\ K by A31,XBOOLE_0:def 4;
CompF(B,G) = A '/\' C '/\' D '/\' E '/\' F by A2,A3,Th32;
then
A33: I /\ K = H or I /\ K misses H by A30,A6,EQREL_1:def 4;
GG=EqClass(u,CompF(A,G)) by A2,A3,Th31;
hence thesis by A29,A26,A32,A28,A33,XBOOLE_0:3;
end;
begin :: Moved from BVFUNC24, AG 19.12.2008
reserve Y for non empty set,
G for Subset of PARTITIONS(Y),
A, B, C, D, E, F, J, M for a_partition of Y,
x,x1,x2,x3,x4,x5,x6,x7,x8,x9 for set;
theorem Th42:
G={A,B,C,D,E,F,J} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & B
<>C & B<>D & B<>E & B<>F & B<>J & C<>D & C<>E & C<>F & C<>J & D<>E & D<>F & D<>
J & E<>F & E<>J & F<>J implies CompF(A,G) = B '/\' C '/\' D '/\' E '/\' F '/\'
J
proof
assume that
A1: G={A,B,C,D,E,F,J} and
A2: A<>B and
A3: A<>C and
A4: A<>D & A<>E and
A5: A<>F & A<>J and
A6: B<>C & B<>D & B<>E & B<>F & B<>J & C<>D & C<>E & C<>F & C<>J & D<>E
& D<>F & D<>J & E<>F & E<>J & F<>J;
A7: G \ {A}={A} \/ {B,C,D,E,F,J} \ {A} by A1,ENUMSET1:16;
( not D in {A})& not E in {A} by A4,TARSKI:def 1;
then
A8: {D,E} \ {A} = {D,E} by ZFMISC_1:63;
A9: ( not F in {A})& not J in {A} by A5,TARSKI:def 1;
A10: not C in {A} by A3,TARSKI:def 1;
A11: not B in {A} by A2,TARSKI:def 1;
{B,C,D,E,F,J} \ {A} = ({B} \/ {C,D,E,F,J}) \ {A} by ENUMSET1:11
.= ({B} \ {A}) \/ ({C,D,E,F,J} \ {A}) by XBOOLE_1:42
.= {B} \/ ({C,D,E,F,J} \ {A}) by A11,ZFMISC_1:59
.= {B} \/ (({C} \/ {D,E,F,J}) \ {A}) by ENUMSET1:7
.= {B} \/ (({C} \ {A}) \/ ({D,E,F,J} \ {A})) by XBOOLE_1:42
.= {B} \/ (({C} \ {A}) \/ (({D,E} \/ {F,J}) \ {A})) by ENUMSET1:5
.= {B} \/ (({C} \ {A}) \/ (({D,E} \ {A}) \/ ({F,J} \ {A}))) by XBOOLE_1:42
.= {B} \/ (({C} \ {A}) \/ ({D,E} \/ {F,J})) by A9,A8,ZFMISC_1:63
.= {B} \/ ({C} \/ ({D,E} \/ {F,J})) by A10,ZFMISC_1:59
.= {B} \/ ({C} \/ {D,E,F,J}) by ENUMSET1:5
.= {B} \/ {C,D,E,F,J} by ENUMSET1:7
.= {B,C,D,E,F,J} by ENUMSET1:11;
then
A12: G \ {A} = {A} \ {A} \/ {B,C,D,E,F,J} by A7,XBOOLE_1:42
.= {} \/ {B,C,D,E,F,J} by XBOOLE_1:37;
A13: '/\' (G \ {A}) c= B '/\' C '/\' D '/\' E '/\' F '/\' J
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume x in '/\' (G \ {A});
then consider h being Function, FF being Subset-Family of Y such that
A14: dom h=(G \ {A}) and
A15: rng h = FF and
A16: for d being set st d in (G \ {A}) holds h.d in d and
A17: x=Intersect FF and
A18: x<>{} by BVFUNC_2:def 1;
A19: C in (G \ {A}) by A12,ENUMSET1:def 4;
then
A20: h.C in C by A16;
set mbcd=(h.B /\ h.C) /\ h.D;
A21: E in (G \ {A}) by A12,ENUMSET1:def 4;
then
A22: h.E in rng h by A14,FUNCT_1:def 3;
set mbc=h.B /\ h.C;
A23: B in (G \ {A}) by A12,ENUMSET1:def 4;
then h.B in B by A16;
then
A24: mbc in INTERSECTION(B,C) by A20,SETFAM_1:def 5;
A25: h.B in rng h by A14,A23,FUNCT_1:def 3;
then
A26: Intersect FF = meet (rng h) by A15,SETFAM_1:def 9;
A27: h.C in rng h by A14,A19,FUNCT_1:def 3;
A28: F in (G \ {A}) by A12,ENUMSET1:def 4;
then
A29: h.F in rng h by A14,FUNCT_1:def 3;
set mbcdef=((h.B /\ h.C) /\ h.D) /\ h.E /\ h.F;
set mbcde=(h.B /\ h.C) /\ h.D /\ h.E;
A30: not x in {{}} by A18,TARSKI:def 1;
A31: J in (G \ {A}) by A12,ENUMSET1:def 4;
then
A32: h.J in rng h by A14,FUNCT_1:def 3;
A33: D in (G \ {A}) by A12,ENUMSET1:def 4;
then
A34: h.D in rng h by A14,FUNCT_1:def 3;
A35: xx c= ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J
proof
let m be object;
assume
A36: m in xx;
then m in h.B & m in h.C by A17,A25,A27,A26,SETFAM_1:def 1;
then
A37: m in h.B /\ h.C by XBOOLE_0:def 4;
m in h.D by A17,A34,A26,A36,SETFAM_1:def 1;
then
A38: m in h.B /\ h.C /\ h.D by A37,XBOOLE_0:def 4;
m in h.E by A17,A22,A26,A36,SETFAM_1:def 1;
then
A39: m in h.B /\ h.C /\ h.D /\ h.E by A38,XBOOLE_0:def 4;
m in h.F by A17,A29,A26,A36,SETFAM_1:def 1;
then
A40: m in h.B /\ h.C /\ h.D /\ h.E /\ h.F by A39,XBOOLE_0:def 4;
m in h. J by A17,A32,A26,A36,SETFAM_1:def 1;
hence thesis by A40,XBOOLE_0:def 4;
end;
then mbcd<>{} by A18;
then
A41: not mbcd in {{}} by TARSKI:def 1;
((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J c= xx
proof
A42: rng h c= {h.B,h.C,h.D,h.E,h.F,h.J}
proof
let u be object;
assume u in rng h;
then consider x1 being object such that
A43: x1 in dom h and
A44: u = h.x1 by FUNCT_1:def 3;
x1=B or x1=C or x1=D or x1=E or x1=F or x1=J by A12,A14,A43,
ENUMSET1:def 4;
hence thesis by A44,ENUMSET1:def 4;
end;
let m be object;
assume
A45: m in ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J;
then
A46: m in h. J by XBOOLE_0:def 4;
A47: m in h.B /\ h.C /\ h.D /\ h.E /\ h.F by A45,XBOOLE_0:def 4;
then
A48: m in h.B /\ h.C /\ h.D /\ h.E by XBOOLE_0:def 4;
then
A49: m in h.E by XBOOLE_0:def 4;
A50: m in h.B /\ h.C /\ h.D by A48,XBOOLE_0:def 4;
then
A51: m in h.D by XBOOLE_0:def 4;
m in h.B /\ h.C by A50,XBOOLE_0:def 4;
then
A52: m in h.B & m in h.C by XBOOLE_0:def 4;
m in h.F by A47,XBOOLE_0:def 4;
then for y being set holds y in rng h implies m in y by A52,A51,A49,A46
,A42,ENUMSET1:def 4;
hence thesis by A17,A25,A26,SETFAM_1:def 1;
end;
then
A53: ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J = x by A35,XBOOLE_0:def 10;
mbc<>{} by A18,A35;
then not mbc in {{}} by TARSKI:def 1;
then mbc in INTERSECTION(B,C) \ {{}} by A24,XBOOLE_0:def 5;
then
A54: mbc in B '/\' C by PARTIT1:def 4;
h.D in D by A16,A33;
then mbcd in INTERSECTION(B '/\' C,D) by A54,SETFAM_1:def 5;
then mbcd in INTERSECTION(B '/\' C,D) \ {{}} by A41,XBOOLE_0:def 5;
then
A55: mbcd in B '/\' C '/\' D by PARTIT1:def 4;
mbcde<>{} by A18,A35;
then
A56: not mbcde in {{}} by TARSKI:def 1;
h.E in E by A16,A21;
then mbcde in INTERSECTION(B '/\' C '/\' D,E) by A55,SETFAM_1:def 5;
then mbcde in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A56,XBOOLE_0:def 5;
then
A57: mbcde in (B '/\' C '/\' D '/\' E) by PARTIT1:def 4;
mbcdef<>{} by A18,A35;
then
A58: not mbcdef in {{}} by TARSKI:def 1;
h.F in F by A16,A28;
then mbcdef in INTERSECTION(B '/\' C '/\' D '/\' E,F) by A57,SETFAM_1:def 5
;
then mbcdef in INTERSECTION(B '/\' C '/\' D '/\' E,F) \ {{}} by A58,
XBOOLE_0:def 5;
then
A59: mbcdef in (B '/\' C '/\' D '/\' E '/\' F) by PARTIT1:def 4;
h.J in J by A16,A31;
then x in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F,J) by A53,A59,
SETFAM_1:def 5;
then x in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F,J) \ {{}} by A30,
XBOOLE_0:def 5;
hence thesis by PARTIT1:def 4;
end;
A60: B '/\' C '/\' D '/\' E '/\' F '/\' J c= '/\' (G \ {A})
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume
A61: x in B '/\' C '/\' D '/\' E '/\' F '/\' J;
then
A62: x<>{} by EQREL_1:def 4;
x in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F,J) \ {{}} by A61,
PARTIT1:def 4;
then consider bcdef,j being set such that
A63: bcdef in B '/\' C '/\' D '/\' E '/\' F and
A64: j in J and
A65: x = bcdef /\ j by SETFAM_1:def 5;
bcdef in INTERSECTION(B '/\' C '/\' D '/\' E,F) \ {{}} by A63,PARTIT1:def 4
;
then consider bcde,f being set such that
A66: bcde in B '/\' C '/\' D '/\' E and
A67: f in F and
A68: bcdef = bcde /\ f by SETFAM_1:def 5;
bcde in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A66,PARTIT1:def 4;
then consider bcd,e being set such that
A69: bcd in B '/\' C '/\' D and
A70: e in E and
A71: bcde = bcd /\ e by SETFAM_1:def 5;
bcd in INTERSECTION(B '/\' C,D) \ {{}} by A69,PARTIT1:def 4;
then consider bc,d being set such that
A72: bc in B '/\' C and
A73: d in D and
A74: bcd = bc /\ d by SETFAM_1:def 5;
bc in INTERSECTION(B,C) \ {{}} by A72,PARTIT1:def 4;
then consider b,c being set such that
A75: b in B & c in C and
A76: bc = b /\ c by SETFAM_1:def 5;
set h = (B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e) +* (F .--> f)
+* (J .--> j);
A77: h.B = b by A6,Th37;
A78: dom h = {J,B,C,D,E,F} by Th38
.= {J} \/ {B,C,D,E,F} by ENUMSET1:11
.= {B,C,D,E,F,J} by ENUMSET1:15;
then D in dom h by ENUMSET1:def 4;
then
A79: h.D in rng h by FUNCT_1:def 3;
A80: for p being set st p in (G \ {A}) holds h.p in p
proof
let p be set;
assume p in (G \ {A});
then p=B or p=C or p=D or p=E or p=F or p=J by A12,ENUMSET1:def 4;
hence thesis by A6,A64,A67,A70,A73,A75,Th37;
end;
E in dom h by A78,ENUMSET1:def 4;
then
A81: h.E in rng h by FUNCT_1:def 3;
C in dom h by A78,ENUMSET1:def 4;
then
A82: h.C in rng h by FUNCT_1:def 3;
A83: h.C = c by A6,Th37;
A84: rng h c= {h.B,h.C,h.D,h.E,h.F,h.J}
proof
let t be object;
assume t in rng h;
then consider x1 being object such that
A85: x1 in dom h and
A86: t = h.x1 by FUNCT_1:def 3;
x1=B or x1=C or x1=D or x1=E or x1=F or x1=J by A78,A85,ENUMSET1:def 4;
hence thesis by A86,ENUMSET1:def 4;
end;
J in dom h by A78,ENUMSET1:def 4;
then
A87: h.J in rng h by FUNCT_1:def 3;
F in dom h by A78,ENUMSET1:def 4;
then
A88: h.F in rng h by FUNCT_1:def 3;
B in dom h by A78,ENUMSET1:def 4;
then
A89: h.B in rng h by FUNCT_1:def 3;
{h.B,h.C,h.D,h.E,h.F,h.J} c= rng h
by A79,A89,A82,A81,A88,A87,ENUMSET1:def 4;
then
A90: rng h = {h.B,h.C,h.D,h.E,h.F,h.J} by A84,XBOOLE_0:def 10;
A91: h.J = j by A6,Th37;
A92: h.F = f by A6,Th37;
A93: h.E = e by A6,Th37;
A94: h.D = d by A6,Th37;
rng h c= bool Y
proof
let t be object;
assume t in rng h;
then t=h.D or t=h.B or t=h.C or t=h.E or t=h.F or t=h.J by A84,
ENUMSET1:def 4;
hence thesis by A64,A67,A70,A73,A75,A94,A77,A83,A93,A92,A91;
end;
then reconsider FF=rng h as Subset-Family of Y;
A95: dom ((B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e) +* (F .-->
f) +* (J .--> j)) = {J,B,C,D,E,F} by Th38
.= {J} \/ {B,C,D,E,F} by ENUMSET1:11
.= {B,C,D,E,F,J} by ENUMSET1:15;
reconsider h as Function;
A96: xx c= Intersect FF
proof
let u be object;
assume
A97: u in xx;
for y be set holds y in FF implies u in y
proof
let y be set;
assume
A98: y in FF;
now
per cases by A84,A98,ENUMSET1:def 4;
case
A99: y=h.D;
u in (d /\ ((b /\ c) /\ e)) /\ f /\ j by A65,A68,A71,A74,A76,A97,
XBOOLE_1:16;
then u in (d /\ ((b /\ c) /\ e /\ f)) /\ j by XBOOLE_1:16;
then u in d /\ (((b /\ c) /\ e /\ f) /\ j) by XBOOLE_1:16;
hence thesis by A94,A99,XBOOLE_0:def 4;
end;
case
A100: y=h.B;
u in (c /\ (d /\ b)) /\ e /\ f /\ j by A65,A68,A71,A74,A76,A97,
XBOOLE_1:16;
then u in c /\ ((d /\ b) /\ e) /\ f /\ j by XBOOLE_1:16;
then u in c /\ ((d /\ e) /\ b) /\ f /\ j by XBOOLE_1:16;
then u in c /\ (((d /\ e) /\ b) /\ f) /\ j by XBOOLE_1:16;
then u in c /\ ((((d /\ e) /\ b) /\ f) /\ j) by XBOOLE_1:16;
then u in c /\ (((d /\ e) /\ (f /\ b)) /\ j) by XBOOLE_1:16;
then u in c /\ ((d /\ e) /\ ((f /\ b) /\ j)) by XBOOLE_1:16;
then u in c /\ ((d /\ e) /\ (f /\ (j /\ b))) by XBOOLE_1:16;
then u in (c /\ (d /\ e)) /\ (f /\ (j /\ b)) by XBOOLE_1:16;
then u in ((c /\ (d /\ e)) /\ f) /\ (j /\ b) by XBOOLE_1:16;
then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ b by XBOOLE_1:16;
hence thesis by A77,A100,XBOOLE_0:def 4;
end;
case
A101: y=h.C;
u in (c /\ (d /\ b)) /\ e /\ f /\ j by A65,A68,A71,A74,A76,A97,
XBOOLE_1:16;
then u in c /\ ((d /\ b) /\ e) /\ f /\ j by XBOOLE_1:16;
then u in c /\ ((d /\ e) /\ b) /\ f /\ j by XBOOLE_1:16;
then u in c /\ (((d /\ e) /\ b) /\ f) /\ j by XBOOLE_1:16;
then u in c /\ ((((d /\ e) /\ b) /\ f) /\ j) by XBOOLE_1:16;
hence thesis by A83,A101,XBOOLE_0:def 4;
end;
case
A102: y=h.E;
u in ((b /\ c) /\ d) /\ (f /\ e) /\ j by A65,A68,A71,A74,A76,A97,
XBOOLE_1:16;
then u in ((b /\ c) /\ d) /\ ((f /\ e) /\ j) by XBOOLE_1:16;
then u in ((b /\ c) /\ d) /\ ((f /\ j) /\ e) by XBOOLE_1:16;
then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ e by XBOOLE_1:16;
hence thesis by A93,A102,XBOOLE_0:def 4;
end;
case
A103: y=h.F;
u in (((b /\ c) /\ d) /\ e) /\ j /\ f by A65,A68,A71,A74,A76,A97,
XBOOLE_1:16;
hence thesis by A92,A103,XBOOLE_0:def 4;
end;
case
y=h.J;
hence thesis by A65,A91,A97,XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u in meet FF by A90,SETFAM_1:def 1;
hence thesis by A90,SETFAM_1:def 9;
end;
A104: Intersect FF = meet (rng h) by A79,SETFAM_1:def 9;
Intersect FF c= xx
proof
let t be object;
assume
A105: t in Intersect FF;
h.C in rng h by A90,ENUMSET1:def 4;
then
A106: t in c by A83,A104,A105,SETFAM_1:def 1;
h.B in rng h by A90,ENUMSET1:def 4;
then t in b by A77,A104,A105,SETFAM_1:def 1;
then
A107: t in b /\ c by A106,XBOOLE_0:def 4;
h.D in rng h by A90,ENUMSET1:def 4;
then t in d by A94,A104,A105,SETFAM_1:def 1;
then
A108: t in (b /\ c) /\ d by A107,XBOOLE_0:def 4;
h.E in rng h by A90,ENUMSET1:def 4;
then t in e by A93,A104,A105,SETFAM_1:def 1;
then
A109: t in (b /\ c) /\ d /\ e by A108,XBOOLE_0:def 4;
h.F in rng h by A90,ENUMSET1:def 4;
then t in f by A92,A104,A105,SETFAM_1:def 1;
then
A110: t in (b /\ c) /\ d /\ e /\ f by A109,XBOOLE_0:def 4;
h.J in rng h by A90,ENUMSET1:def 4;
then t in j by A91,A104,A105,SETFAM_1:def 1;
hence thesis by A65,A68,A71,A74,A76,A110,XBOOLE_0:def 4;
end;
then x = Intersect FF by A96,XBOOLE_0:def 10;
hence thesis by A12,A95,A80,A62,BVFUNC_2:def 1;
end;
CompF(A,G)='/\' (G \ {A}) by BVFUNC_2:def 7;
hence thesis by A60,A13,XBOOLE_0:def 10;
end;
theorem Th43:
G={A,B,C,D,E,F,J} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & B
<>C & B<>D & B<>E & B<>F & B<>J & C<>D & C<>E & C<>F & C<>J & D<>E & D<>F & D<>
J & E<>F & E<>J & F<>J implies CompF(B,G) = A '/\' C '/\' D '/\' E '/\' F '/\'
J
proof
{A,B,C,D,E,F,J}={A,B} \/ {C,D,E,F,J} by ENUMSET1:17
.={B,A,C,D,E,F,J} by ENUMSET1:17;
hence thesis by Th42;
end;
theorem Th44:
G={A,B,C,D,E,F,J} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & B
<>C & B<>D & B<>E & B<>F & B<>J & C<>D & C<>E & C<>F & C<>J & D<>E & D<>F & D<>
J & E<>F & E<>J & F<>J implies CompF(C,G) = A '/\' B '/\' D '/\' E '/\' F '/\'
J
proof
{A,B,C,D,E,F,J}={A,B,C} \/ {D,E,F,J} by ENUMSET1:18
.={A} \/ {B,C} \/ {D,E,F,J} by ENUMSET1:2
.={A,C,B} \/ {D,E,F,J} by ENUMSET1:2
.={A,C} \/ {B} \/ {D,E,F,J} by ENUMSET1:3
.={C,A,B} \/ {D,E,F,J} by ENUMSET1:3
.={C,A,B,D,E,F,J} by ENUMSET1:18;
hence thesis by Th42;
end;
theorem Th45:
G={A,B,C,D,E,F,J} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & B
<>C & B<>D & B<>E & B<>F & B<>J & C<>D & C<>E & C<>F & C<>J & D<>E & D<>F & D<>
J & E<>F & E<>J & F<>J implies CompF(D,G) = A '/\' B '/\' C '/\' E '/\' F '/\'
J
proof
{A,B,C,D,E,F,J}={A,B} \/ {C,D,E,F,J} by ENUMSET1:17
.={A,B} \/ ({C,D} \/ {E,F,J}) by ENUMSET1:8
.={A,B} \/ {D,C,E,F,J} by ENUMSET1:8
.={A,B,D,C,E,F,J} by ENUMSET1:17;
hence thesis by Th44;
end;
theorem Th46:
G={A,B,C,D,E,F,J} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & B
<>C & B<>D & B<>E & B<>F & B<>J & C<>D & C<>E & C<>F & C<>J & D<>E & D<>F & D<>
J & E<>F & E<>J & F<>J implies CompF(E,G) = A '/\' B '/\' C '/\' D '/\' F '/\'
J
proof
{A,B,C,D,E,F,J}={A,B,C} \/ {D,E,F,J} by ENUMSET1:18
.={A,B,C} \/ ({D,E} \/ {F,J}) by ENUMSET1:5
.={A,B,C} \/ {E,D,F,J} by ENUMSET1:5
.={A,B,C,E,D,F,J} by ENUMSET1:18;
hence thesis by Th45;
end;
theorem Th47:
G={A,B,C,D,E,F,J} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & B
<>C & B<>D & B<>E & B<>F & B<>J & C<>D & C<>E & C<>F & C<>J & D<>E & D<>F & D<>
J & E<>F & E<>J & F<>J implies CompF(F,G) = A '/\' B '/\' C '/\' D '/\' E '/\'
J
proof
{A,B,C,D,E,F,J}={A,B,C,D} \/ {E,F,J} by ENUMSET1:19
.={A,B,C,D} \/ {F,E,J} by ENUMSET1:58
.={A,B,C,D,F,E,J} by ENUMSET1:19;
hence thesis by Th46;
end;
theorem
G={A,B,C,D,E,F,J} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & B<>C & B
<>D & B<>E & B<>F & B<>J & C<>D & C<>E & C<>F & C<>J & D<>E & D<>F & D<>J & E<>
F & E<>J & F<>J implies CompF(J,G) = A '/\' B '/\' C '/\' D '/\' E '/\' F
proof
{A,B,C,D,E,F,J}={A,B,C,D,E} \/ {F,J} by ENUMSET1:20
.={A,B,C,D,E,J,F} by ENUMSET1:20;
hence thesis by Th47;
end;
theorem Th49:
for A,B,C,D,E,F,J being set, h being Function, A9,B9,C9,D9,E9,F9
,J9 being set st A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & B<>C & B<>D & B<>E &
B<>F & B<>J & C<>D & C<>E & C<>F & C<>J & D<>E & D<>F & D<>J & E<>F & E<>J & F
<>J & h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F .-->
F9) +* (J .--> J9) +* (A .--> A9) holds h.A = A9 & h.B = B9 & h.C = C9 & h.D =
D9 & h.E = E9 & h.F = F9 & h.J = J9
proof
let A,B,C,D,E,F,J be set;
let h be Function;
let A9,B9,C9,D9,E9,F9,J9 be set;
assume that
A1: A <>B and
A2: A<>C and
A3: A<>D and
A4: A<>E and
A5: A<>F and
A6: A<>J and
A7: B<>C & B<>D & B<>E & B<>F & B<>J & C<>D & C<>E & C<>F & C<>J & D<>E
& D<>F & D<>J & E<>F & E<>J & F<>J and
A8: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (A .--> A9);
A in dom (A .--> A9) by TARSKI:def 1;
then
A10: h.A = (A .--> A9).A by A8,FUNCT_4:13;
not J in dom (A .--> A9) by A6,TARSKI:def 1;
then
A11: h.J=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9)).J by A8,FUNCT_4:11
.= J9 by A7,Th37;
not F in dom (A .--> A9) by A5,TARSKI:def 1;
then
A12: h.F=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9)).F by A8,FUNCT_4:11
.= F9 by A7,Th37;
not E in dom (A .--> A9) by A4,TARSKI:def 1;
then
A13: h.E=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9)).E by A8,FUNCT_4:11
.= E9 by A7,Th37;
not D in dom (A .--> A9) by A3,TARSKI:def 1;
then
A14: h.D=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9)).D by A8,FUNCT_4:11
.= D9 by A7,Th37;
not C in dom (A .--> A9) by A2,TARSKI:def 1;
then
A15: h.C=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9)).C by A8,FUNCT_4:11
.= C9 by A7,Th37;
not B in dom (A .--> A9) by A1,TARSKI:def 1;
then h.B=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9)).B by A8,FUNCT_4:11
.= B9 by A7,Th37;
hence thesis by A10,A15,A14,A13,A12,A11,FUNCOP_1:72;
end;
theorem Th50:
for A,B,C,D,E,F,J being set, h being Function, A9,B9,C9,D9,E9,F9
,J9 being set st h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9)
+* (F .--> F9) +* (J .--> J9) +* (A .--> A9) holds dom h = {A,B,C,D,E,F,J}
proof
let A,B,C,D,E,F,J be set;
let h be Function;
let A9,B9,C9,D9,E9,F9,J9 be set;
assume h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (A .--> A9);
then
dom h = dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +*
(F .--> F9) +* (J .--> J9)) \/ dom (A .--> A9) by FUNCT_4:def 1
.= {J,B,C,D,E,F} \/ dom (A .--> A9) by Th38
.= ({B,C,D,E,F} \/ {J}) \/ {A} by ENUMSET1:11
.= {B,C,D,E,F,J} \/ {A} by ENUMSET1:15
.= {A,B,C,D,E,F,J} by ENUMSET1:16;
hence thesis;
end;
theorem Th51:
for A,B,C,D,E,F,J being set, h being Function, A9,B9,C9,D9,E9,F9
,J9 being set st h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9)
+* (F .--> F9) +* (J .--> J9) +* (A .--> A9) holds rng h = {h.A,h.B,h.C,h.D,h.E
,h.F,h.J}
proof
let A,B,C,D,E,F,J be set;
let h be Function;
let A9,B9,C9,D9,E9,F9,J9 be set;
assume h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (A .--> A9);
then
A1: dom h = {A,B,C,D,E,F,J} by Th50;
then B in dom h by ENUMSET1:def 5;
then
A2: h.B in rng h by FUNCT_1:def 3;
F in dom h by A1,ENUMSET1:def 5;
then
A3: h.F in rng h by FUNCT_1:def 3;
E in dom h by A1,ENUMSET1:def 5;
then
A4: h.E in rng h by FUNCT_1:def 3;
D in dom h by A1,ENUMSET1:def 5;
then
A5: h.D in rng h by FUNCT_1:def 3;
C in dom h by A1,ENUMSET1:def 5;
then
A6: h.C in rng h by FUNCT_1:def 3;
J in dom h by A1,ENUMSET1:def 5;
then
A7: h.J in rng h by FUNCT_1:def 3;
A8: rng h c= {h.A,h.B,h.C,h.D,h.E,h.F,h.J}
proof
let t be object;
assume t in rng h;
then consider x1 being object such that
A9: x1 in dom h and
A10: t = h.x1 by FUNCT_1:def 3;
x1=A or x1=B or x1=C or x1=D or x1=E or x1=F or x1=J by A1,A9,
ENUMSET1:def 5;
hence thesis by A10,ENUMSET1:def 5;
end;
A in dom h by A1,ENUMSET1:def 5;
then
A11: h.A in rng h by FUNCT_1:def 3;
{h.A,h.B,h.C,h.D,h.E,h.F,h.J} c= rng h
by A11,A2,A6,A5,A4,A3,A7,ENUMSET1:def 5;
hence thesis by A8,XBOOLE_0:def 10;
end;
theorem
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J being a_partition
of Y, z,u being Element of Y st G is independent & G={A,B,C,D,E,F,J} & A<>B & A
<>C & A<>D & A<>E & A<>F & A<>J & B<>C & B<>D & B<>E & B<>F & B<>J & C<>D & C<>
E & C<>F & C<>J & D<>E & D<>F & D<>J & E<>F & E<>J & F<>J holds EqClass(u,B
'/\' C '/\' D '/\' E '/\' F '/\' J) meets EqClass(z,A)
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F,J be a_partition of Y;
let z,u be Element of Y;
assume that
A1: G is independent and
A2: G={A,B,C,D,E,F,J} and
A3: A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & B<>C & B<>D & B<>E & B<>F
& B<> J & C<>D & C<>E & C<>F & C<>J & D<>E & D<>F & D<>J & E<>F & E<>J & F<>J;
set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (D .--> EqClass(u,
D)) +* (E .--> EqClass(u,E)) +* (F .--> EqClass(u,F)) +* (J .--> EqClass(u,J))
+* (A .--> EqClass(z,A));
A4: h.A = EqClass(z,A) by A3,Th49;
reconsider GG=EqClass(u,B '/\' C '/\' D '/\' E '/\' F '/\' J) as set;
reconsider I=EqClass(z,A) as set;
GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F) /\ EqClass(u,J) by Th1;
then GG = EqClass(u,B '/\' C '/\' D '/\' E) /\ EqClass(u,F) /\ EqClass(u,J)
by Th1;
then GG = EqClass(u,B '/\' C '/\' D) /\ EqClass(u,E) /\ EqClass(u,F) /\
EqClass(u,J) by Th1;
then
GG = EqClass(u,B '/\' C) /\ EqClass(u,D) /\ EqClass(u,E) /\ EqClass (u,
F) /\ EqClass(u,J) by Th1;
then
A5: GG /\ I = ((((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass
(u,E)) /\ EqClass(u,F) /\ EqClass(u,J)) /\ EqClass(z,A) by Th1;
A6: h.B = EqClass(u,B) by A3,Th49;
A7: h.F = EqClass(u,F) by A3,Th49;
A8: h.E = EqClass(u,E) by A3,Th49;
A9: h.J = EqClass(u,J) by A3,Th49;
A10: h.D = EqClass(u,D) by A3,Th49;
A11: h.C = EqClass(u,C) by A3,Th49;
A12: rng h = {h.A,h.B,h.C,h.D,h.E,h.F,h.J} by Th51;
rng h c= bool Y
proof
let t be object;
assume t in rng h;
then t=h.A or t=h.B or t=h.C or t=h.D or t=h.E or t=h.F or t=h.J by A12,
ENUMSET1:def 5;
hence thesis by A4,A6,A11,A10,A8,A7,A9;
end;
then reconsider FF=rng h as Subset-Family of Y;
A13: dom h = G by A2,Th50;
then A in dom h by A2,ENUMSET1:def 5;
then
A14: h.A in rng h by FUNCT_1:def 3;
then
A15: Intersect FF = meet (rng h) by SETFAM_1:def 9;
for d being set st d in G holds h.d in d
proof
let d be set;
assume d in G;
then d=A or d=B or d=C or d=D or d=E or d=F or d=J by A2,ENUMSET1:def 5;
hence thesis by A4,A6,A11,A10,A8,A7,A9;
end;
then (Intersect FF)<>{} by A1,A13,BVFUNC_2:def 5;
then consider m being object such that
A16: m in Intersect FF by XBOOLE_0:def 1;
C in dom h by A2,A13,ENUMSET1:def 5;
then h.C in rng h by FUNCT_1:def 3;
then
A17: m in EqClass(u,C) by A11,A15,A16,SETFAM_1:def 1;
B in dom h by A2,A13,ENUMSET1:def 5;
then h.B in rng h by FUNCT_1:def 3;
then m in EqClass(u,B) by A6,A15,A16,SETFAM_1:def 1;
then
A18: m in EqClass(u,B) /\ EqClass(u,C) by A17,XBOOLE_0:def 4;
D in dom h by A2,A13,ENUMSET1:def 5;
then h.D in rng h by FUNCT_1:def 3;
then m in EqClass(u,D) by A10,A15,A16,SETFAM_1:def 1;
then
A19: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A18,XBOOLE_0:def 4;
E in dom h by A2,A13,ENUMSET1:def 5;
then h.E in rng h by FUNCT_1:def 3;
then m in EqClass(u,E) by A8,A15,A16,SETFAM_1:def 1;
then
A20: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) by A19,
XBOOLE_0:def 4;
F in dom h by A2,A13,ENUMSET1:def 5;
then h.F in rng h by FUNCT_1:def 3;
then m in EqClass(u,F) by A7,A15,A16,SETFAM_1:def 1;
then
A21: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\
EqClass(u,F) by A20,XBOOLE_0:def 4;
J in dom h by A2,A13,ENUMSET1:def 5;
then h.J in rng h by FUNCT_1:def 3;
then m in EqClass(u,J) by A9,A15,A16,SETFAM_1:def 1;
then
A22: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\
EqClass(u,F) /\ EqClass(u,J) by A21,XBOOLE_0:def 4;
m in EqClass(z,A) by A4,A14,A15,A16,SETFAM_1:def 1;
then
m in EqClass(u,B '/\' C '/\' D '/\' E '/\' F '/\' J) /\ EqClass(z,A) by A5
,A22,XBOOLE_0:def 4;
hence thesis by XBOOLE_0:def 7;
end;
theorem
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J being a_partition
of Y, z,u being Element of Y st G is independent & G={A,B,C,D,E,F,J} & A<>B & A
<>C & A<>D & A<>E & A<>F & A<>J & B<>C & B<>D & B<>E & B<>F & B<>J & C<>D & C<>
E & C<>F & C<>J & D<>E & D<>F & D<>J & E<>F & E<>J & F<>J & EqClass(z,C '/\' D
'/\' E '/\' F '/\' J)= EqClass(u,C '/\' D '/\' E '/\' F '/\' J) holds EqClass(u
,CompF(A,G)) meets EqClass(z,CompF(B,G))
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F,J be a_partition of Y;
let z,u be Element of Y;
assume that
A1: G is independent and
A2: G={A,B,C,D,E,F,J} and
A3: A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & B<>C & B<>D & B<>E & B<>F
& B<> J & C<>D & C<>E & C<>F & C<>J & D<>E & D<>F & D<>J & E<>F & E<>J & F<>J
and
A4: EqClass(z,C '/\' D '/\' E '/\' F '/\' J)= EqClass(u,C '/\' D '/\' E
'/\' F '/\' J);
set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (D .--> EqClass(u,
D)) +* (E .--> EqClass(u,E)) +* (F .--> EqClass(u,F)) +* (J .--> EqClass(u,J))
+* (A .--> EqClass(z,A));
A5: h.A = EqClass(z,A) by A3,Th49;
reconsider L=EqClass(z,C '/\' D '/\' E '/\' F '/\' J) as set;
reconsider GG=EqClass(u,(((B '/\' C) '/\' D) '/\' E '/\' F '/\' J)) as set;
reconsider I=EqClass(z,A) as set;
GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F) /\ EqClass(u,J) by Th1;
then GG = EqClass(u,B '/\' C '/\' D '/\' E) /\ EqClass(u,F) /\ EqClass(u,J)
by Th1;
then GG = EqClass(u,B '/\' C '/\' D) /\ EqClass(u,E) /\ EqClass(u,F) /\
EqClass(u,J) by Th1;
then
GG = EqClass(u,B '/\' C) /\ EqClass(u,D) /\ EqClass(u,E) /\ EqClass (u,
F) /\ EqClass(u,J) by Th1;
then
A6: GG /\ I = ((((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass
(u,E)) /\ EqClass(u,F) /\ EqClass(u,J)) /\ EqClass(z,A) by Th1;
A7: CompF(A,G) = B '/\' C '/\' D '/\' E '/\' F '/\' J by A2,A3,Th42;
reconsider HH=EqClass(z,CompF(B,G)) as set;
A8: z in HH by EQREL_1:def 6;
A9: A '/\' (C '/\' D '/\' E '/\' F '/\' J) = A '/\' (C '/\' D '/\' E '/\'
F) '/\' J by PARTIT1:14
.= A '/\' (C '/\' D '/\' E) '/\' F '/\' J by PARTIT1:14
.= A '/\' (C '/\' D) '/\' E '/\' F '/\' J by PARTIT1:14
.= A '/\' C '/\' D '/\' E '/\' F '/\' J by PARTIT1:14;
A10: h.B = EqClass(u,B) by A3,Th49;
A11: h.F = EqClass(u,F) by A3,Th49;
A12: h.E = EqClass(u,E) by A3,Th49;
A13: h.J = EqClass(u,J) by A3,Th49;
A14: h.D = EqClass(u,D) by A3,Th49;
A15: h.C = EqClass(u,C) by A3,Th49;
A16: rng h = {h.A,h.B,h.C,h.D,h.E,h.F,h.J} by Th51;
rng h c= bool Y
proof
let t be object;
assume t in rng h;
then t=h.A or t=h.B or t=h.C or t=h.D or t=h.E or t=h.F or t=h.J by A16,
ENUMSET1:def 5;
hence thesis by A5,A10,A15,A14,A12,A11,A13;
end;
then reconsider FF=rng h as Subset-Family of Y;
A17: dom h = G by A2,Th50;
then A in dom h by A2,ENUMSET1:def 5;
then
A18: h.A in rng h by FUNCT_1:def 3;
then
A19: Intersect FF = meet (rng h) by SETFAM_1:def 9;
for d being set st d in G holds h.d in d
proof
let d be set;
assume d in G;
then d=A or d=B or d=C or d=D or d=E or d=F or d=J by A2,ENUMSET1:def 5;
hence thesis by A5,A10,A15,A14,A12,A11,A13;
end;
then (Intersect FF)<>{} by A1,A17,BVFUNC_2:def 5;
then consider m being object such that
A20: m in Intersect FF by XBOOLE_0:def 1;
C in dom h by A2,A17,ENUMSET1:def 5;
then h.C in rng h by FUNCT_1:def 3;
then
A21: m in EqClass(u,C) by A15,A19,A20,SETFAM_1:def 1;
B in dom h by A2,A17,ENUMSET1:def 5;
then h.B in rng h by FUNCT_1:def 3;
then m in EqClass(u,B) by A10,A19,A20,SETFAM_1:def 1;
then
A22: m in EqClass(u,B) /\ EqClass(u,C) by A21,XBOOLE_0:def 4;
D in dom h by A2,A17,ENUMSET1:def 5;
then h.D in rng h by FUNCT_1:def 3;
then m in EqClass(u,D) by A14,A19,A20,SETFAM_1:def 1;
then
A23: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A22,XBOOLE_0:def 4;
E in dom h by A2,A17,ENUMSET1:def 5;
then h.E in rng h by FUNCT_1:def 3;
then m in EqClass(u,E) by A12,A19,A20,SETFAM_1:def 1;
then
A24: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) by A23,
XBOOLE_0:def 4;
F in dom h by A2,A17,ENUMSET1:def 5;
then h.F in rng h by FUNCT_1:def 3;
then m in EqClass(u,F) by A11,A19,A20,SETFAM_1:def 1;
then
A25: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\
EqClass(u,F) by A24,XBOOLE_0:def 4;
J in dom h by A2,A17,ENUMSET1:def 5;
then h.J in rng h by FUNCT_1:def 3;
then m in EqClass(u,J) by A13,A19,A20,SETFAM_1:def 1;
then
A26: m in (((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass(u,E)
) /\ EqClass(u,F) /\ EqClass(u,J) by A25,XBOOLE_0:def 4;
m in EqClass(z,A) by A5,A18,A19,A20,SETFAM_1:def 1;
then
A27: GG /\ I <> {} by A6,A26,XBOOLE_0:def 4;
then consider p being object such that
A28: p in GG /\ I by XBOOLE_0:def 1;
GG /\ I in INTERSECTION(A,B '/\' C '/\' D '/\' E '/\' F '/\' J) & not
GG /\ I in {{}} by A27,SETFAM_1:def 5,TARSKI:def 1;
then
GG /\ I in INTERSECTION(A,B '/\' C '/\' D '/\' E '/\' F '/\' J) \ { {}}
by XBOOLE_0:def 5;
then GG /\ I in (A '/\' ((((B '/\' C) '/\' D) '/\' E) '/\' F '/\' J)) by
PARTIT1:def 4;
then reconsider p as Element of Y by A28;
A29: p in GG by A28,XBOOLE_0:def 4;
reconsider K=EqClass(p,C '/\' D '/\' E '/\' F '/\' J) as set;
A30: p in EqClass(p,C '/\' D '/\' E '/\' F '/\' J) by EQREL_1:def 6;
GG = EqClass(u,(((B '/\' (C '/\' D)) '/\' E) '/\' F) '/\' J) by PARTIT1:14;
then GG = EqClass(u,((B '/\' ((C '/\' D) '/\' E)) '/\' F) '/\' J) by
PARTIT1:14;
then GG = EqClass(u,(B '/\' (((C '/\' D) '/\' E) '/\' F)) '/\' J) by
PARTIT1:14;
then GG = EqClass(u,B '/\' (C '/\' D '/\' E '/\' F '/\' J)) by PARTIT1:14;
then GG c= L by A4,BVFUNC11:3;
then K meets L by A29,A30,XBOOLE_0:3;
then K=L by EQREL_1:41;
then
A31: z in K by EQREL_1:def 6;
p in K & p in I by A28,EQREL_1:def 6,XBOOLE_0:def 4;
then
A32: p in I /\ K by XBOOLE_0:def 4;
then
I /\ K in INTERSECTION(A,C '/\' D '/\' E '/\' F '/\' J) & not I /\ K in
{{}} by SETFAM_1:def 5,TARSKI:def 1;
then I /\ K in INTERSECTION(A,C '/\' D '/\' E '/\' F '/\' J) \ {{}} by
XBOOLE_0:def 5;
then
A33: I /\ K in A '/\' (C '/\' D '/\' E '/\' F '/\' J) by PARTIT1:def 4;
z in I by EQREL_1:def 6;
then z in I /\ K by A31,XBOOLE_0:def 4;
then
A34: I /\ K meets HH by A8,XBOOLE_0:3;
CompF(B,G) = A '/\' C '/\' D '/\' E '/\' F '/\' J by A2,A3,Th43;
then p in HH by A32,A33,A34,A9,EQREL_1:def 4;
hence thesis by A7,A29,XBOOLE_0:3;
end;
theorem Th54:
G={A,B,C,D,E,F,J,M} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J &
A<>M & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J & C
<>M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M
implies CompF(A,G) = B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M
proof
assume that
A1: G={A,B,C,D,E,F,J,M} and
A2: A<>B and
A3: A<>C and
A4: A<>D & A<>E and
A5: A<>F & A<>J and
A6: A<>M and
A7: B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J
& C<> M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M;
A8: not B in {A} by A2,TARSKI:def 1;
G \ {A}={A} \/ {B,C,D,E,F,J,M} \ {A} by A1,ENUMSET1:22;
then
A9: G \ {A} = ({A} \ {A}) \/ ({B,C,D,E,F,J,M} \ {A}) by XBOOLE_1:42;
A10: ( not D in {A})& not E in {A} by A4,TARSKI:def 1;
A11: not C in {A} by A3,TARSKI:def 1;
A12: not M in {A} by A6,TARSKI:def 1;
A13: ( not F in {A})& not J in {A} by A5,TARSKI:def 1;
{B,C,D,E,F,J,M} \ {A} =({B} \/ {C,D,E,F,J,M}) \ {A} by ENUMSET1:16
.=({B} \ {A}) \/ ({C,D,E,F,J,M} \ {A}) by XBOOLE_1:42
.={B} \/ ({C,D,E,F,J,M} \ {A}) by A8,ZFMISC_1:59
.={B} \/ (({C} \/ {D,E,F,J,M}) \ {A}) by ENUMSET1:11
.={B} \/ (({C} \ {A}) \/ ({D,E,F,J,M} \ {A})) by XBOOLE_1:42
.={B} \/ (({C} \ {A}) \/ (({D,E} \/ {F,J,M}) \ {A})) by ENUMSET1:8
.={B} \/ (({C} \ {A}) \/ (({D,E} \ {A}) \/ ({F,J,M} \ {A}))) by XBOOLE_1:42
.={B} \/ (({C} \ {A}) \/ ({D,E} \/ ({F,J,M} \ {A}))) by A10,ZFMISC_1:63
.={B} \/ (({C} \ {A}) \/ ({D,E} \/ ({F,J} \/ {M} \ {A}))) by ENUMSET1:3
.={B} \/ (({C} \ {A}) \/ ({D,E} \/ (({F,J} \ {A}) \/ ({M} \ {A})))) by
XBOOLE_1:42
.={B} \/ (({C} \ {A}) \/ ({D,E} \/ ({F,J} \/ ({M} \ {A})))) by A13,
ZFMISC_1:63
.={B} \/ ({C} \/ ({D,E} \/ ({F,J} \/ ({M} \ {A})))) by A11,ZFMISC_1:59
.={B} \/ ({C} \/ ({D,E} \/ ({F,J} \/ {M}))) by A12,ZFMISC_1:59
.={B} \/ ({C} \/ ({D,E} \/ {F,J,M})) by ENUMSET1:3
.={B} \/ ({C} \/ {D,E,F,J,M}) by ENUMSET1:8
.={B} \/ {C,D,E,F,J,M} by ENUMSET1:11
.={B,C,D,E,F,J,M} by ENUMSET1:16;
then
A14: G \ {A} = {} \/ {B,C,D,E,F,J,M} by A9,XBOOLE_1:37;
A15: '/\' (G \ {A}) c= B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume x in '/\' (G \ {A});
then consider h being Function, FF being Subset-Family of Y such that
A16: dom h=(G \ {A}) and
A17: rng h = FF and
A18: for d being set st d in (G \ {A}) holds h.d in d and
A19: x=Intersect FF and
A20: x<>{} by BVFUNC_2:def 1;
A21: C in (G \ {A}) by A14,ENUMSET1:def 5;
then
A22: h.C in C by A18;
set mbcdef=((h.B /\ h.C) /\ h.D) /\ h.E /\ h.F;
set mbcde=(h.B /\ h.C) /\ h.D /\ h.E;
set mbcdefj=((h.B /\ h.C) /\ h.D) /\ h.E /\ h.F /\ h.J;
A23: not x in {{}} by A20,TARSKI:def 1;
A24: J in (G \ {A}) by A14,ENUMSET1:def 5;
then
A25: h.J in rng h by A16,FUNCT_1:def 3;
set mbc=h.B /\ h.C;
A26: B in (G \ {A}) by A14,ENUMSET1:def 5;
then h.B in B by A18;
then
A27: mbc in INTERSECTION(B,C) by A22,SETFAM_1:def 5;
A28: h.B in rng h by A16,A26,FUNCT_1:def 3;
then
A29: Intersect FF = meet (rng h) by A17,SETFAM_1:def 9;
A30: h.C in rng h by A16,A21,FUNCT_1:def 3;
A31: F in (G \ {A}) by A14,ENUMSET1:def 5;
then
A32: h.F in rng h by A16,FUNCT_1:def 3;
set mbcd=(h.B /\ h.C) /\ h.D;
A33: E in (G \ {A}) by A14,ENUMSET1:def 5;
then
A34: h.E in rng h by A16,FUNCT_1:def 3;
A35: M in (G \ {A}) by A14,ENUMSET1:def 5;
then
A36: h.M in rng h by A16,FUNCT_1:def 3;
A37: D in (G \ {A}) by A14,ENUMSET1:def 5;
then
A38: h.D in rng h by A16,FUNCT_1:def 3;
A39: xx c= ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M
proof
let p be object;
assume
A40: p in xx;
then p in h.B & p in h.C by A19,A28,A30,A29,SETFAM_1:def 1;
then
A41: p in h.B /\ h.C by XBOOLE_0:def 4;
p in h.D by A19,A38,A29,A40,SETFAM_1:def 1;
then
A42: p in h.B /\ h.C /\ h.D by A41,XBOOLE_0:def 4;
p in h.E by A19,A34,A29,A40,SETFAM_1:def 1;
then
A43: p in h.B /\ h.C /\ h.D /\ h.E by A42,XBOOLE_0:def 4;
p in h.F by A19,A32,A29,A40,SETFAM_1:def 1;
then
A44: p in h.B /\ h.C /\ h.D /\ h.E /\ h.F by A43,XBOOLE_0:def 4;
p in h.J by A19,A25,A29,A40,SETFAM_1:def 1;
then
A45: p in ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J by A44,XBOOLE_0:def 4
;
p in h.M by A19,A36,A29,A40,SETFAM_1:def 1;
hence thesis by A45,XBOOLE_0:def 4;
end;
then mbcd<>{} by A20;
then
A46: not mbcd in {{}} by TARSKI:def 1;
((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M c= xx
proof
A47: rng h c= {h.B,h.C,h.D,h.E,h.F,h.J,h.M}
proof
let u be object;
assume u in rng h;
then consider x1 being object such that
A48: x1 in dom h and
A49: u = h.x1 by FUNCT_1:def 3;
x1=B or x1=C or x1=D or x1=E or x1=F or x1=J or x1=M by A14,A16,A48,
ENUMSET1:def 5;
hence thesis by A49,ENUMSET1:def 5;
end;
let p be object;
assume
A50: p in ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M;
then
A51: p in h.M by XBOOLE_0:def 4;
A52: p in ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J by A50,XBOOLE_0:def 4
;
then
A53: p in h.J by XBOOLE_0:def 4;
A54: p in h.B /\ h.C /\ h.D /\ h.E /\ h.F by A52,XBOOLE_0:def 4;
then
A55: p in h.B /\ h.C /\ h.D /\ h.E by XBOOLE_0:def 4;
then
A56: p in h.E by XBOOLE_0:def 4;
A57: p in h.B /\ h.C /\ h.D by A55,XBOOLE_0:def 4;
then
A58: p in h.D by XBOOLE_0:def 4;
p in h.B /\ h.C by A57,XBOOLE_0:def 4;
then
A59: p in h.B & p in h.C by XBOOLE_0:def 4;
p in h.F by A54,XBOOLE_0:def 4;
then for y being set holds y in rng h implies p in y by A59,A58,A56,A53
,A51,A47,ENUMSET1:def 5;
hence thesis by A19,A28,A29,SETFAM_1:def 1;
end;
then
A60: ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M = x by A39,
XBOOLE_0:def 10;
mbc<>{} by A20,A39;
then not mbc in {{}} by TARSKI:def 1;
then mbc in INTERSECTION(B,C) \ {{}} by A27,XBOOLE_0:def 5;
then
A61: mbc in B '/\' C by PARTIT1:def 4;
h.D in D by A18,A37;
then mbcd in INTERSECTION(B '/\' C,D) by A61,SETFAM_1:def 5;
then mbcd in INTERSECTION(B '/\' C,D) \ {{}} by A46,XBOOLE_0:def 5;
then
A62: mbcd in B '/\' C '/\' D by PARTIT1:def 4;
mbcde<>{} by A20,A39;
then
A63: not mbcde in {{}} by TARSKI:def 1;
h.E in E by A18,A33;
then mbcde in INTERSECTION(B '/\' C '/\' D,E) by A62,SETFAM_1:def 5;
then mbcde in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A63,XBOOLE_0:def 5;
then
A64: mbcde in (B '/\' C '/\' D '/\' E) by PARTIT1:def 4;
mbcdef<>{} by A20,A39;
then
A65: not mbcdef in {{}} by TARSKI:def 1;
h.F in F by A18,A31;
then mbcdef in INTERSECTION(B '/\' C '/\' D '/\' E,F) by A64,SETFAM_1:def 5
;
then mbcdef in INTERSECTION(B '/\' C '/\' D '/\' E,F) \ {{}} by A65,
XBOOLE_0:def 5;
then
A66: mbcdef in (B '/\' C '/\' D '/\' E '/\' F) by PARTIT1:def 4;
mbcdefj<>{} by A20,A39;
then
A67: not mbcdefj in {{}} by TARSKI:def 1;
h.J in J by A18,A24;
then mbcdefj in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F,J) by A66,
SETFAM_1:def 5;
then mbcdefj in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F,J) \ {{}} by A67
,XBOOLE_0:def 5;
then
A68: mbcdefj in (B '/\' C '/\' D '/\' E '/\' F '/\' J) by PARTIT1:def 4;
h.M in M by A18,A35;
then x in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F '/\' J,M) by A60,A68,
SETFAM_1:def 5;
then x in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F '/\' J,M) \ {{}} by
A23,XBOOLE_0:def 5;
hence thesis by PARTIT1:def 4;
end;
A69: B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M c= '/\' (G \ {A})
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume
A70: x in B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M;
then
A71: x<>{} by EQREL_1:def 4;
x in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F '/\' J,M) \ {{}} by A70,
PARTIT1:def 4;
then consider bcdefj,m being set such that
A72: bcdefj in B '/\' C '/\' D '/\' E '/\' F '/\' J and
A73: m in M and
A74: x = bcdefj /\ m by SETFAM_1:def 5;
bcdefj in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F,J) \ {{}} by A72,
PARTIT1:def 4;
then consider bcdef,j being set such that
A75: bcdef in B '/\' C '/\' D '/\' E '/\' F and
A76: j in J and
A77: bcdefj = bcdef /\ j by SETFAM_1:def 5;
bcdef in INTERSECTION(B '/\' C '/\' D '/\' E,F) \ {{}} by A75,PARTIT1:def 4
;
then consider bcde,f being set such that
A78: bcde in B '/\' C '/\' D '/\' E and
A79: f in F and
A80: bcdef = bcde /\ f by SETFAM_1:def 5;
bcde in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A78,PARTIT1:def 4;
then consider bcd,e being set such that
A81: bcd in B '/\' C '/\' D and
A82: e in E and
A83: bcde = bcd /\ e by SETFAM_1:def 5;
bcd in INTERSECTION(B '/\' C,D) \ {{}} by A81,PARTIT1:def 4;
then consider bc,d being set such that
A84: bc in B '/\' C and
A85: d in D and
A86: bcd = bc /\ d by SETFAM_1:def 5;
bc in INTERSECTION(B,C) \ {{}} by A84,PARTIT1:def 4;
then consider b,c being set such that
A87: b in B & c in C and
A88: bc = b /\ c by SETFAM_1:def 5;
set h = (B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e) +* (F .--> f)
+* (J .--> j) +* (M .--> m);
A89: h.B = b by A7,Th49;
A90: dom ((B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e) +* (F .-->
f) +* (J .--> j) +* (M .--> m)) = {M,B,C,D,E,F,J} by Th50
.= {M} \/ {B,C,D,E,F,J} by ENUMSET1:16
.= {B,C,D,E,F,J,M} by ENUMSET1:21;
then
A91: E in dom h & F in dom h by ENUMSET1:def 5;
A92: D in dom h by A90,ENUMSET1:def 5;
then
A93: h.D in rng h by FUNCT_1:def 3;
A94: J in dom h & M in dom h by A90,ENUMSET1:def 5;
A95: B in dom h & C in dom h by A90,ENUMSET1:def 5;
A96: {h.B,h.C,h.D,h.E,h.F,h.J,h.M} c= rng h
proof
let t be object;
assume t in {h.B,h.C,h.D,h.E,h.F,h.J,h.M};
then t=h.D or t=h.B or t=h.C or t=h.E or t=h.F or t=h.J or t=h.M by
ENUMSET1:def 5;
hence thesis by A92,A95,A91,A94,FUNCT_1:def 3;
end;
A97: for p being set st p in (G \ {A}) holds h.p in p
proof
let p be set;
assume p in (G \ {A});
then p=D or p=B or p=C or p=E or p=F or p=J or p=M by A14,ENUMSET1:def 5;
hence thesis by A7,A73,A76,A79,A82,A85,A87,Th49;
end;
A98: h.C = c by A7,Th49;
A99: h.M = m by A7,Th49;
A100: h.J = j by A7,Th49;
A101: h.F = f by A7,Th49;
A102: rng h c= {h.B,h.C,h.D,h.E,h.F,h.J,h.M}
proof
let t be object;
assume t in rng h;
then consider x1 being object such that
A103: x1 in dom h and
A104: t = h.x1 by FUNCT_1:def 3;
x1=D or x1=B or x1=C or x1=E or x1=F or x1=J or x1=M by A90,A103,
ENUMSET1:def 5;
hence thesis by A104,ENUMSET1:def 5;
end;
then
A105: rng h = {h.B,h.C,h.D,h.E,h.F,h.J,h.M} by A96,XBOOLE_0:def 10;
A106: h.E = e by A7,Th49;
A107: h.D = d by A7,Th49;
rng h c= bool Y
proof
let t be object;
assume t in rng h;
then
t=h.D or t=h.B or t=h.C or t=h.E or t=h.F or t=h.J or t=h.M by A102,
ENUMSET1:def 5;
hence thesis by A73,A76,A79,A82,A85,A87,A107,A89,A98,A106,A101,A100,A99;
end;
then reconsider FF=rng h as Subset-Family of Y;
reconsider h as Function;
A108: xx c= Intersect FF
proof
let u be object;
assume
A109: u in xx;
for y be set holds y in FF implies u in y
proof
let y be set;
assume
A110: y in FF;
now
per cases by A102,A110,ENUMSET1:def 5;
case
A111: y=h.D;
u in (d /\ ((b /\ c) /\ e)) /\ f /\ j /\ m by A74,A77,A80,A83,A86
,A88,A109,XBOOLE_1:16;
then u in (d /\ ((b /\ c) /\ e /\ f)) /\ j /\ m by XBOOLE_1:16;
then u in d /\ (((b /\ c) /\ e /\ f) /\ j) /\ m by XBOOLE_1:16;
then u in d /\ ((((b /\ c) /\ e /\ f) /\ j) /\ m) by XBOOLE_1:16;
hence thesis by A107,A111,XBOOLE_0:def 4;
end;
case
A112: y=h.B;
u in (c /\ (d /\ b)) /\ e /\ f /\ j /\ m by A74,A77,A80,A83,A86,A88
,A109,XBOOLE_1:16;
then u in c /\ ((d /\ b) /\ e) /\ f /\ j /\ m by XBOOLE_1:16;
then u in c /\ ((d /\ e) /\ b) /\ f /\ j /\ m by XBOOLE_1:16;
then u in c /\ (((d /\ e) /\ b) /\ f) /\ j /\ m by XBOOLE_1:16;
then u in c /\ ((((d /\ e) /\ b) /\ f) /\ j) /\ m by XBOOLE_1:16;
then u in c /\ (((d /\ e) /\ (f /\ b)) /\ j) /\ m by XBOOLE_1:16;
then u in c /\ ((d /\ e) /\ ((f /\ b) /\ j)) /\ m by XBOOLE_1:16;
then u in c /\ ((d /\ e) /\ (f /\ (j /\ b))) /\ m by XBOOLE_1:16;
then u in (c /\ (d /\ e)) /\ (f /\ (j /\ b)) /\ m by XBOOLE_1:16;
then u in ((c /\ (d /\ e)) /\ f) /\ (j /\ b) /\ m by XBOOLE_1:16;
then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ b /\ m by XBOOLE_1:16;
then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ m /\ b by XBOOLE_1:16;
hence thesis by A89,A112,XBOOLE_0:def 4;
end;
case
A113: y=h.C;
u in (c /\ (d /\ b)) /\ e /\ f /\ j /\ m by A74,A77,A80,A83,A86,A88
,A109,XBOOLE_1:16;
then u in c /\ ((d /\ b) /\ e) /\ f /\ j /\ m by XBOOLE_1:16;
then u in c /\ ((d /\ e) /\ b) /\ f /\ j /\ m by XBOOLE_1:16;
then u in c /\ (((d /\ e) /\ b) /\ f) /\ j /\ m by XBOOLE_1:16;
then u in c /\ ((((d /\ e) /\ b) /\ f) /\ j) /\ m by XBOOLE_1:16;
then u in c /\ (((((d /\ e) /\ b) /\ f) /\ j) /\ m) by XBOOLE_1:16;
hence thesis by A98,A113,XBOOLE_0:def 4;
end;
case
A114: y=h.E;
u in ((b /\ c) /\ d) /\ (f /\ e) /\ j /\ m by A74,A77,A80,A83,A86
,A88,A109,XBOOLE_1:16;
then u in ((b /\ c) /\ d) /\ ((f /\ e) /\ j) /\ m by XBOOLE_1:16;
then u in ((b /\ c) /\ d) /\ ((f /\ j) /\ e) /\ m by XBOOLE_1:16;
then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ e /\ m by XBOOLE_1:16;
then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ m /\ e by XBOOLE_1:16;
hence thesis by A106,A114,XBOOLE_0:def 4;
end;
case
A115: y=h.F;
u in (((b /\ c) /\ d) /\ e) /\ j /\ f /\ m by A74,A77,A80,A83,A86
,A88,A109,XBOOLE_1:16;
then u in (((b /\ c) /\ d) /\ e) /\ j /\ m /\ f by XBOOLE_1:16;
hence thesis by A101,A115,XBOOLE_0:def 4;
end;
case
A116: y=h.J;
u in (((b /\ c) /\ d) /\ e) /\ f /\ m /\ j by A74,A77,A80,A83,A86
,A88,A109,XBOOLE_1:16;
hence thesis by A100,A116,XBOOLE_0:def 4;
end;
case
y=h.M;
hence thesis by A74,A99,A109,XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u in meet FF by A105,SETFAM_1:def 1;
hence thesis by A105,SETFAM_1:def 9;
end;
A117: Intersect FF = meet (rng h) by A93,SETFAM_1:def 9;
Intersect FF c= xx
proof
let t be object;
assume
A118: t in Intersect FF;
h.C in {h.B,h.C,h.D,h.E,h.F,h.J,h.M} by ENUMSET1:def 5;
then
A119: t in c by A98,A96,A117,A118,SETFAM_1:def 1;
h.B in {h.B,h.C,h.D,h.E,h.F,h.J,h.M} by ENUMSET1:def 5;
then t in b by A89,A96,A117,A118,SETFAM_1:def 1;
then
A120: t in b /\ c by A119,XBOOLE_0:def 4;
h.D in {h.B,h.C,h.D,h.E,h.F,h.J,h.M} by ENUMSET1:def 5;
then t in d by A107,A96,A117,A118,SETFAM_1:def 1;
then
A121: t in (b /\ c) /\ d by A120,XBOOLE_0:def 4;
h.E in {h.B,h.C,h.D,h.E,h.F,h.J,h.M} by ENUMSET1:def 5;
then t in e by A106,A96,A117,A118,SETFAM_1:def 1;
then
A122: t in (b /\ c) /\ d /\ e by A121,XBOOLE_0:def 4;
h.F in {h.B,h.C,h.D,h.E,h.F,h.J,h.M} by ENUMSET1:def 5;
then t in f by A101,A96,A117,A118,SETFAM_1:def 1;
then
A123: t in (b /\ c) /\ d /\ e /\ f by A122,XBOOLE_0:def 4;
h.J in {h.B,h.C,h.D,h.E,h.F,h.J,h.M} by ENUMSET1:def 5;
then t in j by A100,A96,A117,A118,SETFAM_1:def 1;
then
A124: t in (b /\ c) /\ d /\ e /\ f /\ j by A123,XBOOLE_0:def 4;
h.M in {h.B,h.C,h.D,h.E,h.F,h.J,h.M} by ENUMSET1:def 5;
then t in m by A99,A96,A117,A118,SETFAM_1:def 1;
hence thesis by A74,A77,A80,A83,A86,A88,A124,XBOOLE_0:def 4;
end;
then x = Intersect FF by A108,XBOOLE_0:def 10;
hence thesis by A14,A90,A97,A71,BVFUNC_2:def 1;
end;
CompF(A,G)='/\' (G \ {A}) by BVFUNC_2:def 7;
hence thesis by A69,A15,XBOOLE_0:def 10;
end;
theorem Th55:
G={A,B,C,D,E,F,J,M} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J &
A<>M & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J & C
<>M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M
implies CompF(B,G) = A '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M
proof
{A,B,C,D,E,F,J,M}={A,B} \/ {C,D,E,F,J,M} by ENUMSET1:23
.={B,A,C,D,E,F,J,M} by ENUMSET1:23;
hence thesis by Th54;
end;
theorem Th56:
G={A,B,C,D,E,F,J,M} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J &
A<>M & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J & C
<>M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M
implies CompF(C,G) = A '/\' B '/\' D '/\' E '/\' F '/\' J '/\' M
proof
{A,B,C,D,E,F,J,M} ={A,B,C} \/ {D,E,F,J,M} by ENUMSET1:24
.={A} \/ {B,C} \/ {D,E,F,J,M} by ENUMSET1:2
.={A,C,B} \/ {D,E,F,J,M} by ENUMSET1:2
.={A,C,B,D,E,F,J,M} by ENUMSET1:24;
hence thesis by Th55;
end;
theorem Th57:
G={A,B,C,D,E,F,J,M} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J &
A<>M & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J & C
<>M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M
implies CompF(D,G) = A '/\' B '/\' C '/\' E '/\' F '/\' J '/\' M
proof
{A,B,C,D,E,F,J,M} ={A,B} \/ {C,D,E,F,J,M} by ENUMSET1:23
.={A,B} \/ ({C,D} \/ {E,F,J,M}) by ENUMSET1:12
.={A,B} \/ {D,C,E,F,J,M} by ENUMSET1:12
.={A,B,D,C,E,F,J,M} by ENUMSET1:23;
hence thesis by Th56;
end;
theorem Th58:
G={A,B,C,D,E,F,J,M} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J &
A<>M & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J & C
<>M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M
implies CompF(E,G) = A '/\' B '/\' C '/\' D '/\' F '/\' J '/\' M
proof
{A,B,C,D,E,F,J,M} ={A,B,C} \/ {D,E,F,J,M} by ENUMSET1:24
.={A,B,C} \/ ({D,E} \/ {F,J,M}) by ENUMSET1:8
.={A,B,C} \/ ({E,D,F,J,M}) by ENUMSET1:8
.={A,B,C,E,D,F,J,M} by ENUMSET1:24;
hence thesis by Th57;
end;
theorem Th59:
G={A,B,C,D,E,F,J,M} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J &
A<>M & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J & C
<>M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M
implies CompF(F,G) = A '/\' B '/\' C '/\' D '/\' E '/\' J '/\' M
proof
{A,B,C,D,E,F,J,M} ={A,B,C,D} \/ {E,F,J,M} by ENUMSET1:25
.={A,B,C,D} \/ ({E,F} \/ {J,M}) by ENUMSET1:5
.={A,B,C,D} \/ ({F,E,J,M}) by ENUMSET1:5
.={A,B,C,D,F,E,J,M} by ENUMSET1:25;
hence thesis by Th58;
end;
theorem Th60:
G={A,B,C,D,E,F,J,M} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J &
A<>M & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J & C
<>M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M
implies CompF(J,G) = A '/\' B '/\' C '/\' D '/\' E '/\' F '/\' M
proof
{A,B,C,D,E,F,J,M} ={A,B,C,D,E} \/ {F,J,M} by ENUMSET1:26
.={A,B,C,D,E} \/ ({J,F} \/ {M}) by ENUMSET1:3
.={A,B,C,D,E} \/ ({J,F,M}) by ENUMSET1:3
.={A,B,C,D,E,J,F,M} by ENUMSET1:26;
hence thesis by Th59;
end;
theorem
G={A,B,C,D,E,F,J,M} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & A<>M &
B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J & C<>M & D
<>E & D<>F & D<>J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M implies
CompF(M,G) = A '/\' B '/\' C '/\' D '/\' E '/\' F '/\' J
proof
{A,B,C,D,E,F,J,M} ={A,B,C,D,E,F} \/ {J,M} by ENUMSET1:27
.={A,B,C,D,E,F,M,J} by ENUMSET1:27;
hence thesis by Th60;
end;
theorem Th62:
for A,B,C,D,E,F,J,M being set, h being Function, A9,B9,C9,D9,E9,
F9,J9,M9 being set st A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & B<>C & B<>D & B
<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J & C<>M & D<>E & D<>F & D<>
J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M & h = (B .--> B9) +* (C .-->
C9) +* (D .--> D9) +* (E .--> E9) +* (F .--> F9) +* (J .--> J9) +* (M .--> M9)
+* (A .--> A9) holds h.B = B9 & h.C = C9 & h.D = D9 & h.E = E9 & h.F = F9 & h.J
= J9
proof
let A,B,C,D,E,F,J,M be set;
let h be Function;
let A9,B9,C9,D9,E9,F9,J9,M9 be set;
assume that
A1: A<>B and
A2: A<>C and
A3: A<>D and
A4: A<>E and
A5: A<>F and
A6: A<>J and
A7: B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J
& C<> M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M
and
A8: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9) +* (A .--> A9);
not C in dom (A .--> A9) by A2,TARSKI:def 1;
then
A10: h.C=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9)).C by A8,FUNCT_4:11;
not J in dom (A .--> A9) by A6,TARSKI:def 1;
then
A11: h.J=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9)).J by A8,FUNCT_4:11
.= J9 by A7,Th49;
not F in dom (A .--> A9) by A5,TARSKI:def 1;
then
A12: h.F=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9)).F by A8,FUNCT_4:11
.= F9 by A7,Th49;
not E in dom (A .--> A9) by A4,TARSKI:def 1;
then
A13: h.E=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9)).E by A8,FUNCT_4:11
.= E9 by A7,Th49;
not D in dom (A .--> A9) by A3,TARSKI:def 1;
then
A14: h.D=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9)).D by A8,FUNCT_4:11
.= D9 by A7,Th49;
not B in dom (A .--> A9) by A1,TARSKI:def 1;
then h.B=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9)).B by A8,FUNCT_4:11
.= B9 by A7,Th49;
hence thesis by A7,A10,A14,A13,A12,A11,Th49;
end;
theorem Th63:
for A,B,C,D,E,F,J,M being set, h being Function, A9,B9,C9,D9,E9,
F9,J9,M9 being set st h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .-->
E9) +* (F .--> F9) +* (J .--> J9) +* (M .--> M9) +* (A .--> A9) holds dom h = {
A,B,C,D,E,F,J,M}
proof
let A,B,C,D,E,F,J,M be set;
let h be Function;
let A9,B9,C9,D9,E9,F9,J9,M9 be set;
assume
A1: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9) +* (A .--> A9);
A2: dom (A .--> A9) = {A};
dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F .-->
F9) +* (J .--> J9) +* (M .--> M9)) = {M,B,C,D,E,F,J} by Th50
.= {M} \/ {B,C,D,E,F,J} by ENUMSET1:16
.= {B,C,D,E,F,J,M} by ENUMSET1:21;
then
dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F .-->
F9) +* (J .--> J9) +* (M .--> M9) +*(A .--> A9)) = {B,C,D,E,F,J,M} \/ {A} by
A2,FUNCT_4:def 1
.= {A,B,C,D,E,F,J,M} by ENUMSET1:22;
hence thesis by A1;
end;
theorem Th64:
for A,B,C,D,E,F,J,M being set, h being Function, A9,B9,C9,D9,E9,
F9,J9,M9 being set st h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .-->
E9) +* (F .--> F9) +* (J .--> J9) +* (M .--> M9) +* (A .--> A9) holds rng h = {
h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M}
proof
let A,B,C,D,E,F,J,M be set;
let h be Function;
let A9,B9,C9,D9,E9,F9,J9,M9 be set;
assume h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9) +* (A .--> A9);
then
A1: dom h = {A,B,C,D,E,F,J,M} by Th63;
then B in dom h by ENUMSET1:def 6;
then
A2: h.B in rng h by FUNCT_1:def 3;
M in dom h by A1,ENUMSET1:def 6;
then
A3: h.M in rng h by FUNCT_1:def 3;
J in dom h by A1,ENUMSET1:def 6;
then
A4: h.J in rng h by FUNCT_1:def 3;
F in dom h by A1,ENUMSET1:def 6;
then
A5: h.F in rng h by FUNCT_1:def 3;
E in dom h by A1,ENUMSET1:def 6;
then
A6: h.E in rng h by FUNCT_1:def 3;
A7: rng h c= {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M}
proof
let t be object;
assume t in rng h;
then consider x1 being object such that
A8: x1 in dom h and
A9: t = h.x1 by FUNCT_1:def 3;
x1=A or x1=B or x1=C or x1=D or x1=E or x1=F or x1=J or x1=M by A1,A8,
ENUMSET1:def 6;
hence thesis by A9,ENUMSET1:def 6;
end;
D in dom h by A1,ENUMSET1:def 6;
then
A10: h.D in rng h by FUNCT_1:def 3;
C in dom h by A1,ENUMSET1:def 6;
then
A11: h.C in rng h by FUNCT_1:def 3;
A in dom h by A1,ENUMSET1:def 6;
then
A12: h.A in rng h by FUNCT_1:def 3;
{h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M} c= rng h
by A12,A2,A11,A10,A6,A5,A4,A3,ENUMSET1:def 6;
hence thesis by A7,XBOOLE_0:def 10;
end;
theorem
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M being a_partition
of Y, z,u being Element of Y st G is independent & G={A,B,C,D,E,F,J,M} & A<>B &
A<>C & A<>D & A<>E & A<>F & A<>J & A<>M & B<>C & B<>D & B<>E & B<>F & B<>J & B
<>M & C<>D & C<>E & C<>F & C<>J & C<>M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>
J & E<>M & F<>J & F<>M & J<>M holds EqClass(u,B '/\' C '/\' D '/\' E '/\' F
'/\' J '/\' M) /\ EqClass(z,A) <> {}
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F,J,M be a_partition of Y;
let z,u be Element of Y;
assume that
A1: G is independent and
A2: G={A,B,C,D,E,F,J,M} and
A3: A<>B & A<>C & A<>D & A<>E & A<>F & A<>J and
A4: A<>M and
A5: B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J
& C<> M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M;
set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (D .--> EqClass(u,
D)) +* (E .--> EqClass(u,E)) +* (F .--> EqClass(u,F)) +* (J .--> EqClass(u,J))
+* (M .--> EqClass(u,M)) +* (A .--> EqClass(z,A));
A6: h.B = EqClass(u,B) by A3,A5,Th62;
reconsider GG=EqClass(u,B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M) as set;
reconsider I=EqClass(z,A) as set;
GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F '/\' J) /\ EqClass(u,M) by Th1;
then GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F) /\ EqClass(u,J) /\
EqClass(u,M) by Th1;
then
GG = EqClass(u,B '/\' C '/\' D '/\' E) /\ EqClass(u,F) /\ EqClass(u,J)
/\ EqClass(u,M) by Th1;
then GG = (EqClass(u,B '/\' C '/\' D) /\ EqClass(u,E)) /\ EqClass(u,F) /\
EqClass(u,J) /\ EqClass(u,M) by Th1;
then GG = ((((EqClass(u,B '/\' C) /\ EqClass(u,D)) /\ EqClass(u,E)) /\
EqClass(u,F)) /\ EqClass(u,J)) /\ EqClass(u,M) by Th1;
then
A7: GG /\ I = ((((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass
(u,E)) /\ EqClass(u,F) /\ EqClass(u,J) /\ EqClass(u,M)) /\ EqClass(z,A) by
Th1;
A8: h.A = EqClass(z,A) by FUNCT_7:94;
A9: h.C = EqClass(u,C) by A3,A5,Th62;
A10: h.M = EqClass(u,M) by A4,Lm1;
A11: h.J = EqClass(u,J) by A3,A5,Th62;
A12: h.F = EqClass(u,F) by A3,A5,Th62;
A13: h.E = EqClass(u,E) by A3,A5,Th62;
A14: h.D = EqClass(u,D) by A3,A5,Th62;
A15: rng h = {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M} by Th64;
rng h c= bool Y
proof
let t be object;
assume t in rng h;
then t=h.A or t=h.B or t=h.C or t=h.D or t=h.E or t=h.F or t=h.J or t=h.M
by A15,ENUMSET1:def 6;
hence thesis by A8,A6,A9,A14,A13,A12,A11,A10;
end;
then reconsider FF=rng h as Subset-Family of Y;
A16: dom h = G by A2,Th63;
then A in dom h by A2,ENUMSET1:def 6;
then
A17: h.A in rng h by FUNCT_1:def 3;
then
A18: Intersect FF = meet (rng h) by SETFAM_1:def 9;
for d being set st d in G holds h.d in d
proof
let d be set;
assume d in G;
then d=A or d=B or d=C or d=D or d=E or d=F or d=J or d=M by A2,
ENUMSET1:def 6;
hence thesis by A8,A6,A9,A14,A13,A12,A11,A10;
end;
then (Intersect FF)<>{} by A1,A16,BVFUNC_2:def 5;
then consider m being object such that
A19: m in Intersect FF by XBOOLE_0:def 1;
C in dom h by A2,A16,ENUMSET1:def 6;
then h.C in rng h by FUNCT_1:def 3;
then
A20: m in EqClass(u,C) by A9,A18,A19,SETFAM_1:def 1;
B in dom h by A2,A16,ENUMSET1:def 6;
then h.B in rng h by FUNCT_1:def 3;
then m in EqClass(u,B) by A6,A18,A19,SETFAM_1:def 1;
then
A21: m in EqClass(u,B) /\ EqClass(u,C) by A20,XBOOLE_0:def 4;
D in dom h by A2,A16,ENUMSET1:def 6;
then h.D in rng h by FUNCT_1:def 3;
then m in EqClass(u,D) by A14,A18,A19,SETFAM_1:def 1;
then
A22: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A21,XBOOLE_0:def 4;
E in dom h by A2,A16,ENUMSET1:def 6;
then h.E in rng h by FUNCT_1:def 3;
then m in EqClass(u,E) by A13,A18,A19,SETFAM_1:def 1;
then
A23: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) by A22,
XBOOLE_0:def 4;
F in dom h by A2,A16,ENUMSET1:def 6;
then h.F in rng h by FUNCT_1:def 3;
then m in EqClass(u,F) by A12,A18,A19,SETFAM_1:def 1;
then
A24: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\
EqClass(u,F) by A23,XBOOLE_0:def 4;
J in dom h by A2,A16,ENUMSET1:def 6;
then h.J in rng h by FUNCT_1:def 3;
then m in EqClass(u,J) by A11,A18,A19,SETFAM_1:def 1;
then
A25: m in ((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass(u, E)
/\ EqClass(u,F) /\ EqClass(u,J) by A24,XBOOLE_0:def 4;
M in dom h by A2,A16,ENUMSET1:def 6;
then h.M in rng h by FUNCT_1:def 3;
then m in EqClass(u,M) by A10,A18,A19,SETFAM_1:def 1;
then
A26: m in ((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass(u, E)
/\ EqClass(u,F) /\ EqClass(u,J) /\ EqClass(u,M) by A25,XBOOLE_0:def 4;
m in EqClass(z,A) by A8,A17,A18,A19,SETFAM_1:def 1;
hence thesis by A7,A26,XBOOLE_0:def 4;
end;
theorem
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M being a_partition
of Y, z,u being Element of Y st G is independent & G={A,B,C,D,E,F,J,M} & A<>B &
A<>C & A<>D & A<>E & A<>F & A<>J & A<>M & B<>C & B<>D & B<>E & B<>F & B<>J & B
<>M & C<>D & C<>E & C<>F & C<>J & C<>M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>
J & E<>M & F<>J & F<>M & J<>M & EqClass(z,C '/\' D '/\' E '/\' F '/\' J '/\' M)
= EqClass(u,C '/\' D '/\' E '/\' F '/\' J '/\' M) holds EqClass(u,CompF(A,G))
meets EqClass(z,CompF(B,G))
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F,J,M be a_partition of Y;
let z,u be Element of Y;
assume that
A1: G is independent and
A2: G={A,B,C,D,E,F,J,M} and
A3: A<>B & A<>C & A<>D & A<>E & A<>F & A<>J and
A4: A<>M and
A5: B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J
& C<> M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M
and
A6: EqClass(z,C '/\' D '/\' E '/\' F '/\' J '/\' M)= EqClass(u,C '/\' D
'/\' E '/\' F '/\' J '/\' M);
set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (D .--> EqClass(u,
D)) +* (E .--> EqClass(u,E)) +* (F .--> EqClass(u,F)) +* (J .--> EqClass(u,J))
+* (M .--> EqClass(u,M)) +* (A .--> EqClass(z,A));
A7: h.B = EqClass(u,B) by A3,A5,Th62;
set HH=EqClass(z,CompF(B,G)), I=EqClass(z,A), GG=EqClass(u,(((B '/\' C) '/\'
D) '/\' E '/\' F '/\' J '/\' M));
A8: GG=EqClass(u,CompF(A,G)) by A2,A3,A4,A5,Th54;
GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F '/\' J) /\ EqClass(u,M) by Th1;
then GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F) /\ EqClass(u,J) /\
EqClass(u,M) by Th1;
then
GG = EqClass(u,B '/\' C '/\' D '/\' E) /\ EqClass(u,F) /\ EqClass(u,J)
/\ EqClass(u,M) by Th1;
then GG = EqClass(u,B '/\' C '/\' D) /\ EqClass(u,E) /\ EqClass(u,F) /\
EqClass(u,J) /\ EqClass(u,M) by Th1;
then
GG = ((EqClass(u,B '/\' C) /\ EqClass(u,D)) /\ EqClass(u,E)) /\ EqClass
(u,F) /\ EqClass(u,J) /\ EqClass(u,M) by Th1;
then
A9: GG /\ I = ((((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass
(u,E)) /\ EqClass(u,F) /\ EqClass(u,J) /\ EqClass(u,M)) /\ EqClass(z,A) by
Th1;
A10: h.A = EqClass(z,A) by FUNCT_7:94;
A11: h.C = EqClass(u,C) by A3,A5,Th62;
A12: h.M = EqClass(u,M) by A4,Lm1;
A13: h.J = EqClass(u,J) by A3,A5,Th62;
A14: h.F = EqClass(u,F) by A3,A5,Th62;
A15: h.E = EqClass(u,E) by A3,A5,Th62;
A16: h.D = EqClass(u,D) by A3,A5,Th62;
A17: rng h = {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M} by Th64;
rng h c= bool Y
proof
let t be object;
assume t in rng h;
then t=h.A or t=h.B or t=h.C or t=h.D or t=h.E or t=h.F or t=h.J or t=h.M
by A17,ENUMSET1:def 6;
hence thesis by A10,A7,A11,A16,A15,A14,A13,A12;
end;
then reconsider FF=rng h as Subset-Family of Y;
A18: dom h = G by A2,Th63;
then A in dom h by A2,ENUMSET1:def 6;
then
A19: h.A in rng h by FUNCT_1:def 3;
then
A20: Intersect FF = meet (rng h) by SETFAM_1:def 9;
for d being set st d in G holds h.d in d
proof
let d be set;
assume d in G;
then d=A or d=B or d=C or d=D or d=E or d=F or d=J or d=M by A2,
ENUMSET1:def 6;
hence thesis by A10,A7,A11,A16,A15,A14,A13,A12;
end;
then (Intersect FF)<>{} by A1,A18,BVFUNC_2:def 5;
then consider m being object such that
A21: m in Intersect FF by XBOOLE_0:def 1;
C in dom h by A2,A18,ENUMSET1:def 6;
then h.C in rng h by FUNCT_1:def 3;
then
A22: m in EqClass(u,C) by A11,A20,A21,SETFAM_1:def 1;
B in dom h by A2,A18,ENUMSET1:def 6;
then h.B in rng h by FUNCT_1:def 3;
then m in EqClass(u,B) by A7,A20,A21,SETFAM_1:def 1;
then
A23: m in EqClass(u,B) /\ EqClass(u,C) by A22,XBOOLE_0:def 4;
D in dom h by A2,A18,ENUMSET1:def 6;
then h.D in rng h by FUNCT_1:def 3;
then m in EqClass(u,D) by A16,A20,A21,SETFAM_1:def 1;
then
A24: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A23,XBOOLE_0:def 4;
E in dom h by A2,A18,ENUMSET1:def 6;
then h.E in rng h by FUNCT_1:def 3;
then m in EqClass(u,E) by A15,A20,A21,SETFAM_1:def 1;
then
A25: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) by A24,
XBOOLE_0:def 4;
F in dom h by A2,A18,ENUMSET1:def 6;
then h.F in rng h by FUNCT_1:def 3;
then m in EqClass(u,F) by A14,A20,A21,SETFAM_1:def 1;
then
A26: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\
EqClass(u,F) by A25,XBOOLE_0:def 4;
J in dom h by A2,A18,ENUMSET1:def 6;
then h.J in rng h by FUNCT_1:def 3;
then m in EqClass(u,J) by A13,A20,A21,SETFAM_1:def 1;
then
A27: m in (((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass(u,E)
) /\ EqClass(u,F) /\ EqClass(u,J) by A26,XBOOLE_0:def 4;
M in dom h by A2,A18,ENUMSET1:def 6;
then h.M in rng h by FUNCT_1:def 3;
then m in EqClass(u,M) by A12,A20,A21,SETFAM_1:def 1;
then
A28: m in ((((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass(u,E)
) /\ EqClass(u,F) /\ EqClass(u,J)) /\ EqClass(u,M) by A27,XBOOLE_0:def 4;
m in EqClass(z,A) by A10,A19,A20,A21,SETFAM_1:def 1;
then GG /\ I <> {} by A9,A28,XBOOLE_0:def 4;
then consider p being object such that
A29: p in GG /\ I by XBOOLE_0:def 1;
reconsider p as Element of Y by A29;
reconsider K=EqClass(p,C '/\' D '/\' E '/\' F '/\' J '/\' M) as set;
A30: p in GG by A29,XBOOLE_0:def 4;
reconsider L=EqClass(z,C '/\' D '/\' E '/\' F '/\' J '/\' M) as set;
A31: p in EqClass(p,C '/\' D '/\' E '/\' F '/\' J '/\' M) by EQREL_1:def 6;
GG = EqClass(u,(((B '/\' (C '/\' D)) '/\' E) '/\' F) '/\' J '/\' M) by
PARTIT1:14;
then GG = EqClass(u,((B '/\' ((C '/\' D) '/\' E)) '/\' F) '/\' J '/\' M) by
PARTIT1:14;
then GG = EqClass(u,(B '/\' (((C '/\' D) '/\' E) '/\' F)) '/\' J '/\' M) by
PARTIT1:14;
then GG = EqClass(u,B '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) by
PARTIT1:14;
then GG = EqClass(u,B '/\' (C '/\' D '/\' E '/\' F '/\' J '/\' M)) by
PARTIT1:14;
then GG c= L by A6,BVFUNC11:3;
then K meets L by A30,A31,XBOOLE_0:3;
then K=L by EQREL_1:41;
then
A32: z in K by EQREL_1:def 6;
A33: z in HH by EQREL_1:def 6;
z in I by EQREL_1:def 6;
then z in I /\ K by A32,XBOOLE_0:def 4;
then
A34: I /\ K meets HH by A33,XBOOLE_0:3;
A35: A '/\' (C '/\' D '/\' E '/\' F '/\' J '/\' M) = A '/\' (C '/\' D '/\'
E '/\' F '/\' J) '/\' M by PARTIT1:14
.= A '/\' (C '/\' D '/\' E '/\' F) '/\' J '/\' M by PARTIT1:14
.= A '/\' (C '/\' D '/\' E) '/\' F '/\' J '/\' M by PARTIT1:14
.= A '/\' (C '/\' D) '/\' E '/\' F '/\' J '/\' M by PARTIT1:14
.= A '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M by PARTIT1:14;
p in K & p in I by A29,EQREL_1:def 6,XBOOLE_0:def 4;
then
A36: p in I /\ K by XBOOLE_0:def 4;
then I /\ K in INTERSECTION(A,C '/\' D '/\' E '/\' F '/\' J '/\' M) & not I
/\ K in {{}} by SETFAM_1:def 5,TARSKI:def 1;
then I /\ K in INTERSECTION(A,C '/\' D '/\' E '/\' F '/\' J '/\' M) \ { {}}
by XBOOLE_0:def 5;
then
A37: I /\ K in A '/\' (C '/\' D '/\' E '/\' F '/\' J '/\' M) by PARTIT1:def 4;
CompF(B,G) = A '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M by A2,A3,A4,A5,Th55;
then p in HH by A36,A37,A34,A35,EQREL_1:def 4;
hence thesis by A8,A30,XBOOLE_0:3;
end;
Lm3: { x1,x2,x3,x4,x5,x6,x7,x8,x9 } = { x1,x2,x3,x4 } \/ { x5,x6,x7,x8,x9 }
proof
now
let x be object;
A1: x in { x5,x6,x7,x8,x9 } iff x=x5 or x=x6 or x=x7 or x=x8 or x=x9 by
ENUMSET1:def 3;
x in { x1,x2,x3,x4 } iff x=x1 or x=x2 or x=x3 or x=x4 by ENUMSET1:def 2;
hence
x in { x1,x2,x3,x4,x5,x6,x7,x8,x9 } iff x in { x1,x2,x3,x4 } \/ { x5,
x6,x7,x8,x9 } by A1,ENUMSET1:def 7,XBOOLE_0:def 3;
end;
hence thesis by TARSKI:2;
end;
theorem Th67:
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M,N being
a_partition of Y st G={A,B,C,D,E,F,J,M,N} & A<>B & A<>C & A<>D & A<>E & A<>F &
A<>J & A<>M & A<>N & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C
<>E & C<>F & C<>J & C<>M & C<>N & D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>
J & E<>M & E<>N & F<>J & F<>M & F<>N & J<>M & J<>N & M<>N holds CompF(A,G) = B
'/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F,J,M,N be a_partition of Y;
assume that
A1: G={A,B,C,D,E,F,J,M,N} and
A2: A<>B and
A3: A<>C and
A4: A<>D & A<>E and
A5: A<>F & A<>J and
A6: A<>M & A<>N and
A7: B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C<>E & C<>F
& C<> J & C<>M & C<>N & D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>J & E<>M &
E<>N & F<>J & F<>M & F<>N & J<>M & J<>N and
A8: M<>N;
A9: not B in {A} by A2,TARSKI:def 1;
( not D in {A})& not E in {A} by A4,TARSKI:def 1;
then
A10: {D,E} \ {A} = {D,E} by ZFMISC_1:63;
A11: ( not F in {A})& not J in {A} by A5,TARSKI:def 1;
A12: not C in {A} by A3,TARSKI:def 1;
G \ {A}={A} \/ {B,C,D,E,F,J,M,N} \ {A} by A1,ENUMSET1:77;
then
A13: G \ {A} = ({A} \ {A}) \/ ({B,C,D,E,F,J,M,N} \ {A}) by XBOOLE_1:42;
A14: ( not M in {A})& not N in {A} by A6,TARSKI:def 1;
{B,C,D,E,F,J,M,N} \ {A} = ({B} \/ {C,D,E,F,J,M,N}) \ {A} by ENUMSET1:22
.= ({B} \ {A}) \/ ({C,D,E,F,J,M,N} \ {A}) by XBOOLE_1:42
.= {B} \/ ({C,D,E,F,J,M,N} \ {A}) by A9,ZFMISC_1:59
.= {B} \/ (({C} \/ {D,E,F,J,M,N}) \ {A}) by ENUMSET1:16
.= {B} \/ (({C} \ {A}) \/ ({D,E,F,J,M,N} \ {A})) by XBOOLE_1:42
.= {B} \/ (({C} \ {A}) \/ (({D,E} \/ {F,J,M,N}) \ {A})) by ENUMSET1:12
.= {B} \/ (({C} \ {A}) \/ (({D,E} \ {A}) \/ ({F,J,M,N} \ {A}))) by
XBOOLE_1:42
.= {B} \/ (({C} \ {A}) \/ ({D,E} \/ ({F,J} \/ {M,N} \ {A}))) by A10,
ENUMSET1:5
.= {B} \/ (({C} \ {A}) \/ ({D,E} \/ (({F,J} \ {A}) \/ ({M,N} \ {A}))))
by XBOOLE_1:42
.= {B} \/ (({C} \ {A}) \/ ({D,E} \/ ({F,J} \/ ({M,N} \ {A})))) by A11,
ZFMISC_1:63
.= {B} \/ ({C} \/ ({D,E} \/ ({F,J} \/ ({M,N} \ {A})))) by A12,ZFMISC_1:59
.= {B} \/ ({C} \/ ({D,E} \/ ({F,J} \/ {M,N}))) by A14,ZFMISC_1:63
.= {B} \/ ({C} \/ ({D,E} \/ {F,J,M,N})) by ENUMSET1:5
.= {B} \/ ({C} \/ {D,E,F,J,M,N}) by ENUMSET1:12
.= {B} \/ {C,D,E,F,J,M,N} by ENUMSET1:16
.= {B,C,D,E,F,J,M,N} by ENUMSET1:22;
then
A15: G \ {A} = {} \/ {B,C,D,E,F,J,M,N} by A13,XBOOLE_1:37;
A16: '/\' (G \ {A}) c= B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume x in '/\' (G \ {A});
then consider h being Function, FF being Subset-Family of Y such that
A17: dom h=(G \ {A}) and
A18: rng h = FF and
A19: for d being set st d in (G \ {A}) holds h.d in d and
A20: x=Intersect FF and
A21: x<>{} by BVFUNC_2:def 1;
A22: C in (G \ {A}) by A15,ENUMSET1:def 6;
then
A23: h.C in C by A19;
set mbcd=(h.B /\ h.C) /\ h.D;
A24: E in (G \ {A}) by A15,ENUMSET1:def 6;
then
A25: h.E in rng h by A17,FUNCT_1:def 3;
A26: N in (G \ {A}) by A15,ENUMSET1:def 6;
then
A27: h.N in rng h by A17,FUNCT_1:def 3;
set mbc=h.B /\ h.C;
A28: B in (G \ {A}) by A15,ENUMSET1:def 6;
then h.B in B by A19;
then
A29: mbc in INTERSECTION(B,C) by A23,SETFAM_1:def 5;
A30: h.B in rng h by A17,A28,FUNCT_1:def 3;
then
A31: Intersect FF = meet (rng h) by A18,SETFAM_1:def 9;
A32: h.C in rng h by A17,A22,FUNCT_1:def 3;
A33: F in (G \ {A}) by A15,ENUMSET1:def 6;
then
A34: h.F in rng h by A17,FUNCT_1:def 3;
A35: M in (G \ {A}) by A15,ENUMSET1:def 6;
then
A36: h.M in rng h by A17,FUNCT_1:def 3;
A37: J in (G \ {A}) by A15,ENUMSET1:def 6;
then
A38: h.J in rng h by A17,FUNCT_1:def 3;
A39: D in (G \ {A}) by A15,ENUMSET1:def 6;
then
A40: h.D in rng h by A17,FUNCT_1:def 3;
A41: xx c= ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M /\ h.N
proof
let p be object;
assume
A42: p in xx;
then p in h.B & p in h.C by A20,A30,A32,A31,SETFAM_1:def 1;
then
A43: p in h.B /\ h.C by XBOOLE_0:def 4;
p in h.D by A20,A40,A31,A42,SETFAM_1:def 1;
then
A44: p in h.B /\ h.C /\ h.D by A43,XBOOLE_0:def 4;
p in h.E by A20,A25,A31,A42,SETFAM_1:def 1;
then
A45: p in h.B /\ h.C /\ h.D /\ h.E by A44,XBOOLE_0:def 4;
p in h.F by A20,A34,A31,A42,SETFAM_1:def 1;
then
A46: p in h.B /\ h.C /\ h.D /\ h.E /\ h.F by A45,XBOOLE_0:def 4;
p in h.J by A20,A38,A31,A42,SETFAM_1:def 1;
then
A47: p in ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J by A46,XBOOLE_0:def 4
;
p in h.M by A20,A36,A31,A42,SETFAM_1:def 1;
then
A48: p in ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M by A47,
XBOOLE_0:def 4;
p in h.N by A20,A27,A31,A42,SETFAM_1:def 1;
hence thesis by A48,XBOOLE_0:def 4;
end;
then mbcd<>{} by A21;
then
A49: not mbcd in {{}} by TARSKI:def 1;
mbc<>{} by A21,A41;
then not mbc in {{}} by TARSKI:def 1;
then mbc in INTERSECTION(B,C) \ {{}} by A29,XBOOLE_0:def 5;
then
A50: mbc in B '/\' C by PARTIT1:def 4;
h.D in D by A19,A39;
then mbcd in INTERSECTION(B '/\' C,D) by A50,SETFAM_1:def 5;
then mbcd in INTERSECTION(B '/\' C,D) \ {{}} by A49,XBOOLE_0:def 5;
then
A51: mbcd in B '/\' C '/\' D by PARTIT1:def 4;
set mbcdefjm=((h.B /\ h.C) /\ h.D) /\ h.E /\ h.F /\ h.J /\ h.M;
set mbcdefj=((h.B /\ h.C) /\ h.D) /\ h.E /\ h.F /\ h.J;
A52: not x in {{}} by A21,TARSKI:def 1;
set mbcdef=((h.B /\ h.C) /\ h.D) /\ h.E /\ h.F;
set mbcde=(h.B /\ h.C) /\ h.D /\ h.E;
mbcdef<>{} by A21,A41;
then
A53: not mbcdef in {{}} by TARSKI:def 1;
mbcde<>{} by A21,A41;
then
A54: not mbcde in {{}} by TARSKI:def 1;
((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M /\ h.N c= xx
proof
A55: rng h c= {h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N}
proof
let u be object;
assume u in rng h;
then consider x1 being object such that
A56: x1 in dom h and
A57: u = h.x1 by FUNCT_1:def 3;
x1=B or x1=C or x1=D or x1=E or x1=F or x1=J or x1=M or x1=N by A15,A17
,A56,ENUMSET1:def 6;
hence thesis by A57,ENUMSET1:def 6;
end;
let p be object;
assume
A58: p in ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M /\ h.N;
then
A59: p in h.N by XBOOLE_0:def 4;
A60: p in ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M by A58,
XBOOLE_0:def 4;
then
A61: p in ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J by XBOOLE_0:def 4;
then
A62: p in h.J by XBOOLE_0:def 4;
A63: p in h.M by A60,XBOOLE_0:def 4;
A64: p in h.B /\ h.C /\ h.D /\ h.E /\ h.F by A61,XBOOLE_0:def 4;
then
A65: p in h.B /\ h.C /\ h.D /\ h.E by XBOOLE_0:def 4;
then
A66: p in h.E by XBOOLE_0:def 4;
A67: p in h.B /\ h.C /\ h.D by A65,XBOOLE_0:def 4;
then
A68: p in h.D by XBOOLE_0:def 4;
p in h.B /\ h.C by A67,XBOOLE_0:def 4;
then
A69: p in h.B & p in h.C by XBOOLE_0:def 4;
p in h.F by A64,XBOOLE_0:def 4;
then for y being set holds y in rng h implies p in y by A69,A68,A66,A62
,A63,A59,A55,ENUMSET1:def 6;
hence thesis by A20,A30,A31,SETFAM_1:def 1;
end;
then
A70: ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M /\ h.N = x by A41,
XBOOLE_0:def 10;
h.E in E by A19,A24;
then mbcde in INTERSECTION(B '/\' C '/\' D,E) by A51,SETFAM_1:def 5;
then mbcde in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A54,XBOOLE_0:def 5;
then
A71: mbcde in (B '/\' C '/\' D '/\' E) by PARTIT1:def 4;
h.F in F by A19,A33;
then mbcdef in INTERSECTION(B '/\' C '/\' D '/\' E,F) by A71,SETFAM_1:def 5
;
then mbcdef in INTERSECTION(B '/\' C '/\' D '/\' E,F) \ {{}} by A53,
XBOOLE_0:def 5;
then
A72: mbcdef in (B '/\' C '/\' D '/\' E '/\' F) by PARTIT1:def 4;
mbcdefj<>{} by A21,A41;
then
A73: not mbcdefj in {{}} by TARSKI:def 1;
h.J in J by A19,A37;
then mbcdefj in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F,J) by A72,
SETFAM_1:def 5;
then mbcdefj in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F,J) \ {{}} by A73
,XBOOLE_0:def 5;
then
A74: mbcdefj in (B '/\' C '/\' D '/\' E '/\' F '/\' J) by PARTIT1:def 4;
mbcdefjm<>{} by A21,A41;
then
A75: not mbcdefjm in {{}} by TARSKI:def 1;
h.M in M by A19,A35;
then mbcdefjm in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F '/\' J,M) by
A74,SETFAM_1:def 5;
then mbcdefjm in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F '/\' J,M) \ {
{}} by A75,XBOOLE_0:def 5;
then
A76: mbcdefjm in (B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M) by PARTIT1:def 4
;
h.N in N by A19,A26;
then x in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M,N) by
A70,A76,SETFAM_1:def 5;
then x in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M,N) \ {
{}} by A52,XBOOLE_0:def 5;
hence thesis by PARTIT1:def 4;
end;
A77: B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N c= '/\' (G \ {A})
proof
let x be object;
reconsider xx=x as set by TARSKI:1;
assume
A78: x in B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N;
then
A79: x<>{} by EQREL_1:def 4;
x in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M,N) \ {
{}} by A78,PARTIT1:def 4;
then consider bcdefjm,n being set such that
A80: bcdefjm in B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M and
A81: n in N and
A82: x = bcdefjm /\ n by SETFAM_1:def 5;
bcdefjm in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F '/\' J,M) \ {{}
} by A80,PARTIT1:def 4;
then consider bcdefj,m being set such that
A83: bcdefj in B '/\' C '/\' D '/\' E '/\' F '/\' J and
A84: m in M and
A85: bcdefjm = bcdefj /\ m by SETFAM_1:def 5;
bcdefj in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F,J) \ {{}} by A83,
PARTIT1:def 4;
then consider bcdef,j being set such that
A86: bcdef in B '/\' C '/\' D '/\' E '/\' F and
A87: j in J and
A88: bcdefj = bcdef /\ j by SETFAM_1:def 5;
bcdef in INTERSECTION(B '/\' C '/\' D '/\' E,F) \ {{}} by A86,PARTIT1:def 4
;
then consider bcde,f being set such that
A89: bcde in B '/\' C '/\' D '/\' E and
A90: f in F and
A91: bcdef = bcde /\ f by SETFAM_1:def 5;
bcde in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A89,PARTIT1:def 4;
then consider bcd,e being set such that
A92: bcd in B '/\' C '/\' D and
A93: e in E and
A94: bcde = bcd /\ e by SETFAM_1:def 5;
bcd in INTERSECTION(B '/\' C,D) \ {{}} by A92,PARTIT1:def 4;
then consider bc,d being set such that
A95: bc in B '/\' C and
A96: d in D and
A97: bcd = bc /\ d by SETFAM_1:def 5;
bc in INTERSECTION(B,C) \ {{}} by A95,PARTIT1:def 4;
then consider b,c being set such that
A98: b in B and
A99: c in C and
A100: bc = b /\ c by SETFAM_1:def 5;
set h = (B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e) +* (F .--> f)
+* (J .--> j) +* (M .--> m) +* (N .--> n);
A101: h.N = n by FUNCT_7:94;
A102: dom ((B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e) +* (F .-->
f) +* (J .--> j) +* (M .--> m) +* (N .--> n)) = {N,B,C,D,E,F,J,M} by Th63
.= {N} \/ {B,C,D,E,F,J,M} by ENUMSET1:22
.= {B,C,D,E,F,J,M,N} by ENUMSET1:28;
then
A103: C in dom h by ENUMSET1:def 6;
A104: for p being set st p in (G \ {A}) holds h.p in p
proof
let p be set;
assume p in (G \ {A});
then p=B or p=C or p=D or p=E or p=F or p=J or p=M or p=N by A15,
ENUMSET1:def 6;
hence thesis by A7,A8,A81,A84,A87,A90,A93,A96,A98,A99,Lm1,Th62,FUNCT_7:94
;
end;
A105: D in dom h by A102,ENUMSET1:def 6;
then
A106: h.D in rng h by FUNCT_1:def 3;
A107: N in dom h by A102,ENUMSET1:def 6;
A108: M in dom h by A102,ENUMSET1:def 6;
A109: J in dom h by A102,ENUMSET1:def 6;
A110: F in dom h by A102,ENUMSET1:def 6;
A111: h.B = b by A7,Th62;
A112: rng h c= {h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N}
proof
let t be object;
assume t in rng h;
then consider x1 being object such that
A113: x1 in dom h and
A114: t = h.x1 by FUNCT_1:def 3;
now
per cases by A102,A113,ENUMSET1:def 6;
case
x1=D;
hence thesis by A114,ENUMSET1:def 6;
end;
case
x1=B;
hence thesis by A114,ENUMSET1:def 6;
end;
case
x1=C;
hence thesis by A114,ENUMSET1:def 6;
end;
case
x1=E;
hence thesis by A114,ENUMSET1:def 6;
end;
case
x1=F;
hence thesis by A114,ENUMSET1:def 6;
end;
case
x1=J;
hence thesis by A114,ENUMSET1:def 6;
end;
case
x1=M;
hence thesis by A114,ENUMSET1:def 6;
end;
case
x1=N;
hence thesis by A114,ENUMSET1:def 6;
end;
end;
hence thesis;
end;
A115: h.J = j by A7,Th62;
A116: h.F = f by A7,Th62;
A117: h.M = m by A8,Lm1;
A118: h.E = e by A7,Th62;
A119: h.C = c by A7,Th62;
A120: h.D = d by A7,Th62;
rng h c= bool Y
proof
let t be object;
assume
A121: t in rng h;
now
per cases by A112,A121,ENUMSET1:def 6;
case
t=h.D;
hence thesis by A96,A120;
end;
case
t=h.B;
hence thesis by A98,A111;
end;
case
t=h.C;
hence thesis by A99,A119;
end;
case
t=h.E;
hence thesis by A93,A118;
end;
case
t=h.F;
hence thesis by A90,A116;
end;
case
t=h.J;
hence thesis by A87,A115;
end;
case
t=h.M;
hence thesis by A84,A117;
end;
case
t=h.N;
hence thesis by A81,A101;
end;
end;
hence thesis;
end;
then reconsider FF=rng h as Subset-Family of Y;
A122: E in dom h by A102,ENUMSET1:def 6;
A123: B in dom h by A102,ENUMSET1:def 6;
{h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N} c= rng h
proof
let t be object;
assume
A124: t in {h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N};
now
per cases by A124,ENUMSET1:def 6;
case
t=h.D;
hence thesis by A105,FUNCT_1:def 3;
end;
case
t=h.B;
hence thesis by A123,FUNCT_1:def 3;
end;
case
t=h.C;
hence thesis by A103,FUNCT_1:def 3;
end;
case
t=h.E;
hence thesis by A122,FUNCT_1:def 3;
end;
case
t=h.F;
hence thesis by A110,FUNCT_1:def 3;
end;
case
t=h.J;
hence thesis by A109,FUNCT_1:def 3;
end;
case
t=h.M;
hence thesis by A108,FUNCT_1:def 3;
end;
case
t=h.N;
hence thesis by A107,FUNCT_1:def 3;
end;
end;
hence thesis;
end;
then
A125: rng h = {h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N} by A112,XBOOLE_0:def 10;
reconsider h as Function;
A126: xx c= Intersect FF
proof
let u be object;
assume
A127: u in xx;
for y be set holds y in FF implies u in y
proof
let y be set;
assume
A128: y in FF;
now
per cases by A112,A128,ENUMSET1:def 6;
case
A129: y=h.D;
u in (d /\ ((b /\ c) /\ e)) /\ f /\ j /\ m /\ n by A82,A85,A88,A91
,A94,A97,A100,A127,XBOOLE_1:16;
then u in (d /\ ((b /\ c) /\ e /\ f)) /\ j /\ m /\ n by XBOOLE_1:16
;
then u in d /\ (((b /\ c) /\ e /\ f) /\ j) /\ m /\ n by XBOOLE_1:16
;
then u in d /\ ((((b /\ c) /\ e /\ f) /\ j) /\ m) /\ n by
XBOOLE_1:16;
then u in d /\ ((((b /\ c) /\ e /\ f) /\ j) /\ m /\ n) by
XBOOLE_1:16;
hence thesis by A120,A129,XBOOLE_0:def 4;
end;
case
A130: y=h.B;
u in (c /\ (d /\ b)) /\ e /\ f /\ j /\ m /\ n by A82,A85,A88,A91
,A94,A97,A100,A127,XBOOLE_1:16;
then u in c /\ ((d /\ b) /\ e) /\ f /\ j /\ m /\ n by XBOOLE_1:16;
then u in c /\ ((d /\ e) /\ b) /\ f /\ j /\ m /\ n by XBOOLE_1:16;
then u in c /\ (((d /\ e) /\ b) /\ f) /\ j /\ m /\ n by XBOOLE_1:16
;
then u in c /\ ((((d /\ e) /\ b) /\ f) /\ j) /\ m /\ n by
XBOOLE_1:16;
then u in c /\ (((d /\ e) /\ (f /\ b)) /\ j) /\ m /\ n by
XBOOLE_1:16;
then u in c /\ ((d /\ e) /\ ((f /\ b) /\ j)) /\ m /\ n by
XBOOLE_1:16;
then u in c /\ ((d /\ e) /\ (f /\ (j /\ b))) /\ m /\ n by
XBOOLE_1:16;
then u in (c /\ (d /\ e)) /\ (f /\ (j /\ b)) /\ m /\ n by
XBOOLE_1:16;
then u in ((c /\ (d /\ e)) /\ f) /\ (j /\ b) /\ m /\ n by
XBOOLE_1:16;
then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ b /\ m /\ n by
XBOOLE_1:16;
then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ (m /\ b) /\ n by
XBOOLE_1:16;
then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ (m /\ b /\ n) by
XBOOLE_1:16;
then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ (m /\ (b /\ n)) by
XBOOLE_1:16;
then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ m /\ (n /\ b) by
XBOOLE_1:16;
then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ m /\ n /\ b by
XBOOLE_1:16;
hence thesis by A111,A130,XBOOLE_0:def 4;
end;
case
A131: y=h.C;
u in (c /\ (d /\ b)) /\ e /\ f /\ j /\ m /\ n by A82,A85,A88,A91
,A94,A97,A100,A127,XBOOLE_1:16;
then u in c /\ ((d /\ b) /\ e) /\ f /\ j /\ m /\ n by XBOOLE_1:16;
then u in c /\ ((d /\ e) /\ b) /\ f /\ j /\ m /\ n by XBOOLE_1:16;
then u in c /\ (((d /\ e) /\ b) /\ f) /\ j /\ m /\ n by XBOOLE_1:16
;
then u in c /\ ((((d /\ e) /\ b) /\ f) /\ j) /\ m /\ n by
XBOOLE_1:16;
then u in c /\ (((((d /\ e) /\ b) /\ f) /\ j) /\ m) /\ n by
XBOOLE_1:16;
then u in c /\ ((((((d /\ e) /\ b) /\ f) /\ j) /\ m) /\ n) by
XBOOLE_1:16;
hence thesis by A119,A131,XBOOLE_0:def 4;
end;
case
A132: y=h.E;
u in ((b /\ c) /\ d) /\ (f /\ e) /\ j /\ m /\ n by A82,A85,A88,A91
,A94,A97,A100,A127,XBOOLE_1:16;
then u in ((b /\ c) /\ d) /\ ((f /\ e) /\ j) /\ m /\ n by
XBOOLE_1:16;
then u in ((b /\ c) /\ d) /\ ((f /\ j) /\ e) /\ m /\ n by
XBOOLE_1:16;
then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ e /\ m /\ n by
XBOOLE_1:16;
then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ (e /\ m) /\ n by
XBOOLE_1:16;
then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ ((m /\ e) /\ n) by
XBOOLE_1:16;
then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ (m /\ (n /\ e)) by
XBOOLE_1:16;
then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ m /\ (n /\ e) by
XBOOLE_1:16;
then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ m /\ n /\ e by
XBOOLE_1:16;
hence thesis by A118,A132,XBOOLE_0:def 4;
end;
case
A133: y=h.F;
u in (((b /\ c) /\ d) /\ e) /\ j /\ f /\ m /\ n by A82,A85,A88,A91
,A94,A97,A100,A127,XBOOLE_1:16;
then u in (((b /\ c) /\ d) /\ e) /\ j /\ m /\ f /\ n by XBOOLE_1:16
;
then u in (((b /\ c) /\ d) /\ e) /\ j /\ m /\ n /\ f by XBOOLE_1:16
;
hence thesis by A116,A133,XBOOLE_0:def 4;
end;
case
A134: y=h.J;
u in (((b /\ c) /\ d) /\ e) /\ f /\ m /\ j /\ n by A82,A85,A88,A91
,A94,A97,A100,A127,XBOOLE_1:16;
then u in (((b /\ c) /\ d) /\ e) /\ f /\ m /\ n /\ j by XBOOLE_1:16
;
hence thesis by A115,A134,XBOOLE_0:def 4;
end;
case
A135: y=h.M;
u in (((b /\ c) /\ d) /\ e) /\ f /\ j /\ n /\ m by A82,A85,A88,A91
,A94,A97,A100,A127,XBOOLE_1:16;
hence thesis by A117,A135,XBOOLE_0:def 4;
end;
case
y=h.N;
hence thesis by A82,A101,A127,XBOOLE_0:def 4;
end;
end;
hence thesis;
end;
then u in meet FF by A125,SETFAM_1:def 1;
hence thesis by A125,SETFAM_1:def 9;
end;
A136: Intersect FF = meet (rng h) by A106,SETFAM_1:def 9;
Intersect FF c= xx
proof
let t be object;
assume
A137: t in Intersect FF;
h.C in rng h by A125,ENUMSET1:def 6;
then
A138: t in c by A119,A136,A137,SETFAM_1:def 1;
h.B in rng h by A125,ENUMSET1:def 6;
then t in b by A111,A136,A137,SETFAM_1:def 1;
then
A139: t in b /\ c by A138,XBOOLE_0:def 4;
h.D in rng h by A125,ENUMSET1:def 6;
then t in d by A120,A136,A137,SETFAM_1:def 1;
then
A140: t in (b /\ c) /\ d by A139,XBOOLE_0:def 4;
h.E in rng h by A125,ENUMSET1:def 6;
then t in e by A118,A136,A137,SETFAM_1:def 1;
then
A141: t in (b /\ c) /\ d /\ e by A140,XBOOLE_0:def 4;
h.F in rng h by A125,ENUMSET1:def 6;
then t in f by A116,A136,A137,SETFAM_1:def 1;
then
A142: t in (b /\ c) /\ d /\ e /\ f by A141,XBOOLE_0:def 4;
h.J in rng h by A125,ENUMSET1:def 6;
then t in j by A115,A136,A137,SETFAM_1:def 1;
then
A143: t in (b /\ c) /\ d /\ e /\ f /\ j by A142,XBOOLE_0:def 4;
h.M in rng h by A125,ENUMSET1:def 6;
then t in m by A117,A136,A137,SETFAM_1:def 1;
then
A144: t in (b /\ c) /\ d /\ e /\ f /\ j /\ m by A143,XBOOLE_0:def 4;
h.N in rng h by A125,ENUMSET1:def 6;
then t in n by A101,A136,A137,SETFAM_1:def 1;
hence thesis by A82,A85,A88,A91,A94,A97,A100,A144,XBOOLE_0:def 4;
end;
then x = Intersect FF by A126,XBOOLE_0:def 10;
hence thesis by A15,A102,A104,A79,BVFUNC_2:def 1;
end;
CompF(A,G)='/\' (G \ {A}) by BVFUNC_2:def 7;
hence thesis by A77,A16,XBOOLE_0:def 10;
end;
theorem Th68:
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M,N being
a_partition of Y st G={A,B,C,D,E,F,J,M,N} & A<>B & A<>C & A<>D & A<>E & A<>F &
A<>J & A<>M & A<>N & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C
<>E & C<>F & C<>J & C<>M & C<>N & D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>
J & E<>M & E<>N & F<>J & F<>M & F<>N & J<>M & J<>N & M<>N holds CompF(B,G) = A
'/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F,J,M,N be a_partition of Y;
{A,B,C,D,E,F,J,M,N} ={A,B} \/ {C,D,E,F,J,M,N} by ENUMSET1:78
.={B,A,C,D,E,F,J,M,N} by ENUMSET1:78;
hence thesis by Th67;
end;
theorem Th69:
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M,N being
a_partition of Y st G={A,B,C,D,E,F,J,M,N} & A<>B & A<>C & A<>D & A<>E & A<>F &
A<>J & A<>M & A<>N & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C
<>E & C<>F & C<>J & C<>M & C<>N & D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>
J & E<>M & E<>N & F<>J & F<>M & F<>N & J<>M & J<>N & M<>N holds CompF(C,G) = A
'/\' B '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F,J,M,N be a_partition of Y;
{A,B,C,D,E,F,J,M,N} ={A,B,C} \/ {D,E,F,J,M,N} by ENUMSET1:79
.={A} \/ {B,C} \/ {D,E,F,J,M,N} by ENUMSET1:2
.={A,C,B} \/ {D,E,F,J,M,N} by ENUMSET1:2
.={A,C,B,D,E,F,J,M,N} by ENUMSET1:79;
hence thesis by Th68;
end;
theorem Th70:
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M,N being
a_partition of Y st G={A,B,C,D,E,F,J,M,N} & A<>B & A<>C & A<>D & A<>E & A<>F &
A<>J & A<>M & A<>N & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C
<>E & C<>F & C<>J & C<>M & C<>N & D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>
J & E<>M & E<>N & F<>J & F<>M & F<>N & J<>M & J<>N & M<>N holds CompF(D,G) = A
'/\' B '/\' C '/\' E '/\' F '/\' J '/\' M '/\' N
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F,J,M,N be a_partition of Y;
{A,B,C,D,E,F,J,M,N} ={A,B} \/ {C,D,E,F,J,M,N} by ENUMSET1:78
.={A,B} \/ ({C,D} \/ {E,F,J,M,N}) by ENUMSET1:17
.={A,B} \/ {D,C,E,F,J,M,N} by ENUMSET1:17
.={A,B,D,C,E,F,J,M,N} by ENUMSET1:78;
hence thesis by Th69;
end;
theorem Th71:
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M,N being
a_partition of Y st G={A,B,C,D,E,F,J,M,N} & A<>B & A<>C & A<>D & A<>E & A<>F &
A<>J & A<>M & A<>N & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C
<>E & C<>F & C<>J & C<>M & C<>N & D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>
J & E<>M & E<>N & F<>J & F<>M & F<>N & J<>M & J<>N & M<>N holds CompF(E,G) = A
'/\' B '/\' C '/\' D '/\' F '/\' J '/\' M '/\' N
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F,J,M,N be a_partition of Y;
{A,B,C,D,E,F,J,M,N} ={A,B,C} \/ {D,E,F,J,M,N} by ENUMSET1:79
.={A,B,C} \/ ({D,E} \/ {F,J,M,N}) by ENUMSET1:12
.={A,B,C} \/ ({E,D,F,J,M,N}) by ENUMSET1:12
.={A,B,C,E,D,F,J,M,N} by ENUMSET1:79;
hence thesis by Th70;
end;
theorem Th72:
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M,N being
a_partition of Y st G={A,B,C,D,E,F,J,M,N} & A<>B & A<>C & A<>D & A<>E & A<>F &
A<>J & A<>M & A<>N & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C
<>E & C<>F & C<>J & C<>M & C<>N & D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>
J & E<>M & E<>N & F<>J & F<>M & F<>N & J<>M & J<>N & M<>N holds CompF(F,G) = A
'/\' B '/\' C '/\' D '/\' E '/\' J '/\' M '/\' N
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F,J,M,N be a_partition of Y;
{A,B,C,D,E,F,J,M,N} ={A,B,C,D} \/ {E,F,J,M,N} by Lm3
.={A,B,C,D} \/ ({E,F} \/ {J,M,N}) by ENUMSET1:8
.={A,B,C,D} \/ ({F,E,J,M,N}) by ENUMSET1:8
.={A,B,C,D,F,E,J,M,N} by Lm3;
hence thesis by Th71;
end;
theorem Th73:
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M,N being
a_partition of Y st G={A,B,C,D,E,F,J,M,N} & A<>B & A<>C & A<>D & A<>E & A<>F &
A<>J & A<>M & A<>N & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C
<>E & C<>F & C<>J & C<>M & C<>N & D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>
J & E<>M & E<>N & F<>J & F<>M & F<>N & J<>M & J<>N & M<>N holds CompF(J,G) = A
'/\' B '/\' C '/\' D '/\' E '/\' F '/\' M '/\' N
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F,J,M,N be a_partition of Y;
{A,B,C,D,E,F,J,M,N} ={A,B,C,D,E} \/ {F,J,M,N} by ENUMSET1:81
.={A,B,C,D,E} \/ ({J,F} \/ {M,N}) by ENUMSET1:5
.={A,B,C,D,E} \/ ({J,F,M,N}) by ENUMSET1:5
.={A,B,C,D,E,J,F,M,N} by ENUMSET1:81;
hence thesis by Th72;
end;
theorem Th74:
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M,N being
a_partition of Y st G={A,B,C,D,E,F,J,M,N} & A<>B & A<>C & A<>D & A<>E & A<>F &
A<>J & A<>M & A<>N & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C
<>E & C<>F & C<>J & C<>M & C<>N & D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>
J & E<>M & E<>N & F<>J & F<>M & F<>N & J<>M & J<>N & M<>N holds CompF(M,G) = A
'/\' B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' N
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F,J,M,N be a_partition of Y;
{A,B,C,D,E,F,J,M,N} ={A,B,C,D,E,F} \/ {J,M,N} by ENUMSET1:82
.={A,B,C,D,E,F} \/ ({J,M} \/ {N}) by ENUMSET1:3
.={A,B,C,D,E,F} \/ {M,J,N} by ENUMSET1:3
.={A,B,C,D,E,F,M,J,N} by ENUMSET1:82;
hence thesis by Th73;
end;
theorem
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M,N being
a_partition of Y st G={A,B,C,D,E,F,J,M,N} & A<>B & A<>C & A<>D & A<>E & A<>F &
A<>J & A<>M & A<>N & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C
<>E & C<>F & C<>J & C<>M & C<>N & D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>
J & E<>M & E<>N & F<>J & F<>M & F<>N & J<>M & J<>N & M<>N holds CompF(N,G) = A
'/\' B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F,J,M,N be a_partition of Y;
{A,B,C,D,E,F,J,M,N} ={A,B,C,D,E,F,J} \/ {M,N} by ENUMSET1:83
.={A,B,C,D,E,F,J,N,M} by ENUMSET1:83;
hence thesis by Th74;
end;
theorem Th76:
for A,B,C,D,E,F,J,M,N being set, h being Function, A9,B9,C9,D9,
E9,F9,J9,M9,N9 being set st A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & A<>M & A
<>N & B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C<>E & C<>F & C<>
J & C<>M & C<>N & D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>J & E<>M & E<>N
& F<>J & F<>M & F<>N & J<>M & J<>N & M<>N & h = (B .--> B9) +* (C .--> C9) +* (
D .--> D9) +* (E .--> E9) +* (F .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N
.--> N9) +* (A .--> A9) holds h.A = A9 & h.B = B9 & h.C = C9 & h.D = D9 & h.E =
E9 & h.F = F9 & h.J = J9 & h.M = M9 & h.N = N9
proof
let A,B,C,D,E,F,J,M,N be set;
let h be Function;
let A9,B9,C9,D9,E9,F9,J9,M9,N9 be set;
assume that
A1: A<>B and
A2: A<>C and
A3: A<>D and
A4: A<>E and
A5: A<>F and
A6: A<>J and
A7: A<>M and
A8: A<>N and
A9: B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C<>E & C<>F
& C<> J & C<>M & C<>N & D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>J & E<>M &
E<>N & F<>J & F<>M & F<>N & J<>M & J<>N and
A10: M<>N and
A11: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9) +* (A .--> A9);
A in dom (A .--> A9) by TARSKI:def 1;
then
A13: h.A = (A .--> A9).A by A11,FUNCT_4:13;
not E in dom (A .--> A9) by A4,TARSKI:def 1;
then
A14: h.E=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).E by A11,FUNCT_4:11
.= E9 by A9,Th62;
not N in dom (A .--> A9) by A8,TARSKI:def 1;
then
A15: h.N=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).N by A11,FUNCT_4:11
.= N9 by FUNCT_7:94;
not D in dom (A .--> A9) by A3,TARSKI:def 1;
then
A16: h.D=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).D by A11,FUNCT_4:11
.= D9 by A9,Th62;
not C in dom (A .--> A9) by A2,TARSKI:def 1;
then
A17: h.C=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).C by A11,FUNCT_4:11;
not J in dom (A .--> A9) by A6,TARSKI:def 1;
then
A18: h.J=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).J by A11,FUNCT_4:11
.= J9 by A9,Th62;
not F in dom (A .--> A9) by A5,TARSKI:def 1;
then
A19: h.F=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).F by A11,FUNCT_4:11
.= F9 by A9,Th62;
not M in dom (A .--> A9) by A7,TARSKI:def 1;
then
A20: h.M=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).M by A11,FUNCT_4:11
.= M9 by A10,Lm1;
not B in dom (A .--> A9) by A1,TARSKI:def 1;
then h.B=((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)).B by A11,FUNCT_4:11
.= B9 by A9,Th62;
hence thesis by A9,A13,A17,A16,A14,A19,A18,A20,A15,Th62,FUNCOP_1:72;
end;
theorem Th77:
for A,B,C,D,E,F,J,M,N being set, h being Function, A9,B9,C9,D9,
E9,F9,J9,M9,N9 being set st h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E
.--> E9) +* (F .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9) +* (A .-->
A9) holds dom h = {A,B,C,D,E,F,J,M,N}
proof
let A,B,C,D,E,F,J,M,N be set;
let h be Function;
let A9,B9,C9,D9,E9,F9,J9,M9,N9 be set;
assume
A1: h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9) +* (A .--> A9);
A2: dom (A .--> A9) = {A};
dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F .-->
F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9)) = {N,B,C,D,E,F,J,M} by Th63
.= {N} \/ {B,C,D,E,F,J,M} by ENUMSET1:22
.= {B,C,D,E,F,J,M,N} by ENUMSET1:28;
then
dom ((B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F .-->
F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9) +*(A .--> A9)) = {B,C,D,E,F,J,
M,N} \/ {A} by A2,FUNCT_4:def 1
.= {A,B,C,D,E,F,J,M,N} by ENUMSET1:77;
hence thesis by A1;
end;
theorem Th78:
for A,B,C,D,E,F,J,M,N being set, h being Function, A9,B9,C9,D9,
E9,F9,J9,M9,N9 being set st h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E
.--> E9) +* (F .--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9) +* (A .-->
A9) holds rng h = {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N}
proof
let A,B,C,D,E,F,J,M,N be set;
let h be Function;
let A9,B9,C9,D9,E9,F9,J9,M9,N9 be set;
assume h = (B .--> B9) +* (C .--> C9) +* (D .--> D9) +* (E .--> E9) +* (F
.--> F9) +* (J .--> J9) +* (M .--> M9) +* (N .--> N9) +* (A .--> A9);
then
A1: dom h = {A,B,C,D,E,F,J,M,N} by Th77;
then
A2: B in dom h by ENUMSET1:def 7;
A3: M in dom h by A1,ENUMSET1:def 7;
A4: J in dom h by A1,ENUMSET1:def 7;
A5: N in dom h by A1,ENUMSET1:def 7;
A6: D in dom h by A1,ENUMSET1:def 7;
A7: C in dom h by A1,ENUMSET1:def 7;
A8: rng h c= {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N}
proof
let t be object;
assume t in rng h;
then consider x1 being object such that
A9: x1 in dom h and
A10: t = h.x1 by FUNCT_1:def 3;
now
per cases by A1,A9,ENUMSET1:def 7;
case
x1=A;
hence thesis by A10,ENUMSET1:def 7;
end;
case
x1=B;
hence thesis by A10,ENUMSET1:def 7;
end;
case
x1=C;
hence thesis by A10,ENUMSET1:def 7;
end;
case
x1=D;
hence thesis by A10,ENUMSET1:def 7;
end;
case
x1=E;
hence thesis by A10,ENUMSET1:def 7;
end;
case
x1=F;
hence thesis by A10,ENUMSET1:def 7;
end;
case
x1=J;
hence thesis by A10,ENUMSET1:def 7;
end;
case
x1=M;
hence thesis by A10,ENUMSET1:def 7;
end;
case
x1=N;
hence thesis by A10,ENUMSET1:def 7;
end;
end;
hence thesis;
end;
A11: F in dom h by A1,ENUMSET1:def 7;
A12: E in dom h by A1,ENUMSET1:def 7;
A13: A in dom h by A1,ENUMSET1:def 7;
{h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N} c= rng h
proof
let t be object;
assume
A14: t in {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N};
now
per cases by A14,ENUMSET1:def 7;
case
t=h.A;
hence thesis by A13,FUNCT_1:def 3;
end;
case
t=h.B;
hence thesis by A2,FUNCT_1:def 3;
end;
case
t=h.C;
hence thesis by A7,FUNCT_1:def 3;
end;
case
t=h.D;
hence thesis by A6,FUNCT_1:def 3;
end;
case
t=h.E;
hence thesis by A12,FUNCT_1:def 3;
end;
case
t=h.F;
hence thesis by A11,FUNCT_1:def 3;
end;
case
t=h.J;
hence thesis by A4,FUNCT_1:def 3;
end;
case
t=h.M;
hence thesis by A3,FUNCT_1:def 3;
end;
case
t=h.N;
hence thesis by A5,FUNCT_1:def 3;
end;
end;
hence thesis;
end;
hence thesis by A8,XBOOLE_0:def 10;
end;
theorem
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M,N being
a_partition of Y, z,u being Element of Y st G is independent & G={A,B,C,D,E,F,J
,M,N} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & A<>M & A<>N & B<>C & B<>D & B
<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C<>E & C<>F & C<>J & C<>M & C<>N & D<>
E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>J & E<>M & E<>N & F<>J & F<>M & F<>N
& J<>M & J<>N & M<>N holds EqClass(u,B '/\' C '/\' D '/\' E '/\' F '/\' J '/\'
M '/\' N) /\ EqClass(z,A) <> {}
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F,J,M,N be a_partition of Y;
let z,u be Element of Y;
assume that
A1: G is independent and
A2: G={A,B,C,D,E,F,J,M,N} and
A3: A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & A<>M & A<>N & B<>C & B<>D
& B<> E & B<>F & B<>J & B<>M & B<>N & C<>D & C<>E & C<>F & C<>J & C<>M & C<>N &
D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>J & E<>M & E<>N & F<>J & F<>M & F
<>N & J<>M & J<>N & M<>N;
set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (D .--> EqClass(u,
D)) +* (E .--> EqClass(u,E)) +* (F .--> EqClass(u,F)) +* (J .--> EqClass(u,J))
+* (M .--> EqClass(u,M)) +* (N .--> EqClass(u,N)) +* (A .--> EqClass(z,A));
A4: h.A = EqClass(z,A) by A3,Th76;
set GG=EqClass(u,B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N);
GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M) /\ EqClass(
u,N) by Th1;
then
GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F '/\' J) /\ EqClass(u,M) /\
EqClass(u,N) by Th1;
then GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F) /\ EqClass(u,J) /\
EqClass(u,M) /\ EqClass(u,N) by Th1;
then
GG = EqClass(u,B '/\' C '/\' D '/\' E) /\ EqClass(u,F) /\ EqClass(u,J)
/\ EqClass(u,M) /\ EqClass(u,N) by Th1;
then GG = (EqClass(u,B '/\' C '/\' D) /\ EqClass(u,E)) /\ EqClass(u,F) /\
EqClass(u,J) /\ EqClass(u,M) /\ EqClass(u,N) by Th1;
then GG = ((((EqClass(u,B '/\' C) /\ EqClass(u,D)) /\ EqClass(u,E)) /\
EqClass(u,F)) /\ EqClass(u,J)) /\ EqClass(u,M) /\ EqClass(u,N) by Th1;
then
A5: GG /\ EqClass(z,A) = ((((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u, D)
) /\ EqClass(u,E)) /\ EqClass(u,F) /\ EqClass(u,J) /\ EqClass(u,M) /\ EqClass(u
,N)) /\ EqClass(z,A) by Th1;
A6: h.B = EqClass(u,B) by A3,Th76;
A7: h.F = EqClass(u,F) by A3,Th76;
A8: h.E = EqClass(u,E) by A3,Th76;
A9: h.M = EqClass(u,M) by A3,Th76;
A10: h.J = EqClass(u,J) by A3,Th76;
A11: h.N = EqClass(u,N) by A3,Th76;
A12: h.D = EqClass(u,D) by A3,Th76;
A13: h.C = EqClass(u,C) by A3,Th76;
A14: rng h = {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N} by Th78;
rng h c= bool Y
proof
let t be object;
assume t in rng h;
then t=h.A or t=h.B or t=h.C or t=h.D or t=h.E or t=h.F or t=h.J or t=h.M
or t=h.N by A14,ENUMSET1:def 7;
hence thesis by A4,A6,A13,A12,A8,A7,A10,A9,A11;
end;
then reconsider FF=rng h as Subset-Family of Y;
A15: dom h = G by A2,Th77;
then A in dom h by A2,ENUMSET1:def 7;
then
A16: h.A in rng h by FUNCT_1:def 3;
then
A17: Intersect FF = meet (rng h) by SETFAM_1:def 9;
for d being set st d in G holds h.d in d
proof
let d be set;
assume d in G;
then d=A or d=B or d=C or d=D or d=E or d=F or d=J or d=M or d=N by A2,
ENUMSET1:def 7;
hence thesis by A4,A6,A13,A12,A8,A7,A10,A9,A11;
end;
then (Intersect FF)<>{} by A1,A15,BVFUNC_2:def 5;
then consider m being object such that
A18: m in Intersect FF by XBOOLE_0:def 1;
C in dom h by A2,A15,ENUMSET1:def 7;
then h.C in rng h by FUNCT_1:def 3;
then
A19: m in EqClass(u,C) by A13,A17,A18,SETFAM_1:def 1;
B in dom h by A2,A15,ENUMSET1:def 7;
then h.B in rng h by FUNCT_1:def 3;
then m in EqClass(u,B) by A6,A17,A18,SETFAM_1:def 1;
then
A20: m in EqClass(u,B) /\ EqClass(u,C) by A19,XBOOLE_0:def 4;
D in dom h by A2,A15,ENUMSET1:def 7;
then h.D in rng h by FUNCT_1:def 3;
then m in EqClass(u,D) by A12,A17,A18,SETFAM_1:def 1;
then
A21: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A20,XBOOLE_0:def 4;
E in dom h by A2,A15,ENUMSET1:def 7;
then h.E in rng h by FUNCT_1:def 3;
then m in EqClass(u,E) by A8,A17,A18,SETFAM_1:def 1;
then
A22: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) by A21,
XBOOLE_0:def 4;
F in dom h by A2,A15,ENUMSET1:def 7;
then h.F in rng h by FUNCT_1:def 3;
then m in EqClass(u,F) by A7,A17,A18,SETFAM_1:def 1;
then
A23: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\
EqClass(u,F) by A22,XBOOLE_0:def 4;
J in dom h by A2,A15,ENUMSET1:def 7;
then h.J in rng h by FUNCT_1:def 3;
then m in EqClass(u,J) by A10,A17,A18,SETFAM_1:def 1;
then
A24: m in ((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass(u, E)
/\ EqClass(u,F) /\ EqClass(u,J) by A23,XBOOLE_0:def 4;
M in dom h by A2,A15,ENUMSET1:def 7;
then h.M in rng h by FUNCT_1:def 3;
then m in EqClass(u,M) by A9,A17,A18,SETFAM_1:def 1;
then
A25: m in ((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass(u, E)
/\ EqClass(u,F) /\ EqClass(u,J) /\ EqClass(u,M) by A24,XBOOLE_0:def 4;
N in dom h by A2,A15,ENUMSET1:def 7;
then h.N in rng h by FUNCT_1:def 3;
then m in EqClass(u,N) by A11,A17,A18,SETFAM_1:def 1;
then
A26: m in ((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass(u, E)
/\ EqClass(u,F) /\ EqClass(u,J) /\ EqClass(u,M) /\ EqClass(u,N) by A25,
XBOOLE_0:def 4;
m in EqClass(z,A) by A4,A16,A17,A18,SETFAM_1:def 1;
hence thesis by A5,A26,XBOOLE_0:def 4;
end;
theorem
for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M,N being
a_partition of Y, z,u being Element of Y st G is independent & G={A,B,C,D,E,F,J
,M,N} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & A<>M & A<>N & B<>C & B<>D & B
<>E & B<>F & B<>J & B<>M & B<>N & C<>D & C<>E & C<>F & C<>J & C<>M & C<>N & D<>
E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>J & E<>M & E<>N & F<>J & F<>M & F<>N
& J<>M & J<>N & M<>N & EqClass(z,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N)=
EqClass(u,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N) holds EqClass(u,CompF(A,
G)) meets EqClass(z,CompF(B,G))
proof
let G be Subset of PARTITIONS(Y);
let A,B,C,D,E,F,J,M,N be a_partition of Y;
let z,u be Element of Y;
assume that
A1: G is independent and
A2: G={A,B,C,D,E,F,J,M,N} and
A3: A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & A<>M & A<>N & B<>C & B<>D
& B<> E & B<>F & B<>J & B<>M & B<>N & C<>D & C<>E & C<>F & C<>J & C<>M & C<>N &
D<>E & D<>F & D<>J & D<>M & D<>N & E<>F & E<>J & E<>M & E<>N & F<>J & F<>M & F
<>N & J<>M & J<>N & M<>N and
A4: EqClass(z,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N)= EqClass(u,C
'/\' D '/\' E '/\' F '/\' J '/\' M '/\' N);
set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (D .--> EqClass(u,
D)) +* (E .--> EqClass(u,E)) +* (F .--> EqClass(u,F)) +* (J .--> EqClass(u,J))
+* (M .--> EqClass(u,M)) +* (N .--> EqClass(u,N)) +* (A .--> EqClass(z,A));
A5: h.A = EqClass(z,A) by A3,Th76;
set L=EqClass(z,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N);
set GG=EqClass(u,(((B '/\' C) '/\' D) '/\' E '/\' F '/\' J '/\' M '/\' N));
reconsider I=EqClass(z,A) as set;
GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M) /\ EqClass(
u,N) by Th1;
then GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F '/\' J) /\ EqClass(u,M )
/\ EqClass(u,N) by Th1;
then GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F) /\ EqClass(u,J) /\
EqClass(u,M) /\ EqClass(u,N) by Th1;
then
GG = EqClass(u,B '/\' C '/\' D '/\' E) /\ EqClass(u,F) /\ EqClass(u,J)
/\ EqClass(u,M) /\ EqClass(u,N) by Th1;
then GG = (((EqClass(u,B '/\' C '/\' D)) /\ EqClass(u,E)) /\ EqClass(u,F ))
/\ EqClass(u,J) /\ EqClass(u,M) /\ EqClass(u,N) by Th1;
then
GG = ((EqClass(u,B '/\' C) /\ EqClass(u,D)) /\ EqClass(u,E)) /\ EqClass
(u,F) /\ EqClass(u,J) /\ EqClass(u,M) /\ EqClass(u,N) by Th1;
then
A6: GG /\ I = ((((((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\
EqClass(u,E)) /\ EqClass(u,F)) /\ EqClass(u,J)) /\ EqClass(u,M) /\ EqClass(u,N)
) /\ EqClass(z,A) by Th1;
A7: CompF(A,G) = B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N by A2,A3
,Th67;
reconsider HH=EqClass(z,CompF(B,G)) as set;
A8: z in HH by EQREL_1:def 6;
A9: A '/\' (C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N) = A '/\' (C '/\'
D '/\' E '/\' F '/\' J '/\' M) '/\' N by PARTIT1:14
.= A '/\' (C '/\' D '/\' E '/\' F '/\' J) '/\' M '/\' N by PARTIT1:14
.= A '/\' (C '/\' D '/\' E '/\' F) '/\' J '/\' M '/\' N by PARTIT1:14
.= A '/\' (C '/\' D '/\' E) '/\' F '/\' J '/\' M '/\' N by PARTIT1:14
.= A '/\' (C '/\' D) '/\' E '/\' F '/\' J '/\' M '/\' N by PARTIT1:14
.= A '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N by PARTIT1:14;
A10: h.B = EqClass(u,B) by A3,Th76;
A11: h.N = EqClass(u,N) by A3,Th76;
A12: h.D = EqClass(u,D) by A3,Th76;
A13: h.C = EqClass(u,C) by A3,Th76;
A14: h.M = EqClass(u,M) by A3,Th76;
A15: h.J = EqClass(u,J) by A3,Th76;
A16: h.F = EqClass(u,F) by A3,Th76;
A17: h.E = EqClass(u,E) by A3,Th76;
A18: rng h = {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N} by Th78;
rng h c= bool Y
proof
let t be object;
assume t in rng h;
then t=h.A or t=h.B or t=h.C or t=h.D or t=h.E or t=h.F or t=h.J or t=h.M
or t=h.N by A18,ENUMSET1:def 7;
hence thesis by A5,A10,A13,A12,A17,A16,A15,A14,A11;
end;
then reconsider FF=rng h as Subset-Family of Y;
A19: dom h = G by A2,Th77;
then A in dom h by A2,ENUMSET1:def 7;
then
A20: h.A in rng h by FUNCT_1:def 3;
then
A21: Intersect FF = meet (rng h) by SETFAM_1:def 9;
for d being set st d in G holds h.d in d
proof
let d be set;
assume d in G;
then d=A or d=B or d=C or d=D or d=E or d=F or d=J or d=M or d=N by A2,
ENUMSET1:def 7;
hence thesis by A5,A10,A13,A12,A17,A16,A15,A14,A11;
end;
then (Intersect FF)<>{} by A1,A19,BVFUNC_2:def 5;
then consider m being object such that
A22: m in Intersect FF by XBOOLE_0:def 1;
C in dom h by A2,A19,ENUMSET1:def 7;
then h.C in rng h by FUNCT_1:def 3;
then
A23: m in EqClass(u,C) by A13,A21,A22,SETFAM_1:def 1;
B in dom h by A2,A19,ENUMSET1:def 7;
then h.B in rng h by FUNCT_1:def 3;
then m in EqClass(u,B) by A10,A21,A22,SETFAM_1:def 1;
then
A24: m in EqClass(u,B) /\ EqClass(u,C) by A23,XBOOLE_0:def 4;
D in dom h by A2,A19,ENUMSET1:def 7;
then h.D in rng h by FUNCT_1:def 3;
then m in EqClass(u,D) by A12,A21,A22,SETFAM_1:def 1;
then
A25: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A24,XBOOLE_0:def 4;
E in dom h by A2,A19,ENUMSET1:def 7;
then h.E in rng h by FUNCT_1:def 3;
then m in EqClass(u,E) by A17,A21,A22,SETFAM_1:def 1;
then
A26: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) by A25,
XBOOLE_0:def 4;
F in dom h by A2,A19,ENUMSET1:def 7;
then h.F in rng h by FUNCT_1:def 3;
then m in EqClass(u,F) by A16,A21,A22,SETFAM_1:def 1;
then
A27: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\
EqClass(u,F) by A26,XBOOLE_0:def 4;
J in dom h by A2,A19,ENUMSET1:def 7;
then h.J in rng h by FUNCT_1:def 3;
then m in EqClass(u,J) by A15,A21,A22,SETFAM_1:def 1;
then
A28: m in (((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass(u,E
)) /\ EqClass(u,F) /\ EqClass(u,J) by A27,XBOOLE_0:def 4;
M in dom h by A2,A19,ENUMSET1:def 7;
then h.M in rng h by FUNCT_1:def 3;
then m in EqClass(u,M) by A14,A21,A22,SETFAM_1:def 1;
then
A29: m in ((((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass(u,E
)) /\ EqClass(u,F) /\ EqClass(u,J)) /\ EqClass(u,M) by A28,XBOOLE_0:def 4;
N in dom h by A2,A19,ENUMSET1:def 7;
then h.N in rng h by FUNCT_1:def 3;
then m in EqClass(u,N) by A11,A21,A22,SETFAM_1:def 1;
then
A30: m in ((((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass(u,E
)) /\ EqClass(u,F) /\ EqClass(u,J)) /\ EqClass(u,M) /\ EqClass(u,N) by A29,
XBOOLE_0:def 4;
m in EqClass(z,A) by A5,A20,A21,A22,SETFAM_1:def 1;
then GG /\ I <> {} by A6,A30,XBOOLE_0:def 4;
then consider p being object such that
A31: p in GG /\ I by XBOOLE_0:def 1;
reconsider p as Element of Y by A31;
set K=EqClass(p,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N);
A32: p in GG by A31,XBOOLE_0:def 4;
A33: p in EqClass(p,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N) by
EQREL_1:def 6;
GG=EqClass(u,(((B '/\' (C '/\' D)) '/\' E) '/\' F) '/\' J '/\' M '/\'
N) by PARTIT1:14;
then GG=EqClass(u,((B '/\' ((C '/\' D) '/\' E)) '/\' F) '/\' J '/\' M '/\'
N) by PARTIT1:14;
then GG=EqClass(u,(B '/\' (((C '/\' D) '/\' E) '/\' F)) '/\' J '/\' M '/\'
N) by PARTIT1:14;
then GG=EqClass(u,B '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M '/\'
N) by PARTIT1:14;
then GG= EqClass(u,B '/\' (((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)
'/\' N) by PARTIT1:14;
then GG=EqClass(u,B '/\' (C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N)) by
PARTIT1:14;
then GG c= L by A4,BVFUNC11:3;
then K meets L by A32,A33,XBOOLE_0:3;
then K=L by EQREL_1:41;
then
A34: z in K by EQREL_1:def 6;
p in K & p in I by A31,EQREL_1:def 6,XBOOLE_0:def 4;
then
A35: p in I /\ K by XBOOLE_0:def 4;
then I /\ K in INTERSECTION(A,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N)
& not I /\ K in {{}} by SETFAM_1:def 5,TARSKI:def 1;
then
A36: I /\ K in INTERSECTION(A,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N)
\ {{}} by XBOOLE_0:def 5;
z in I by EQREL_1:def 6;
then z in I /\ K by A34,XBOOLE_0:def 4;
then
A37: I /\ K meets HH by A8,XBOOLE_0:3;
CompF(B,G) = A '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N by A2,A3,Th68
;
then I /\ K in CompF(B,G) by A36,A9,PARTIT1:def 4;
then p in HH by A35,A37,EQREL_1:def 4;
hence thesis by A7,A32,XBOOLE_0:3;
end;