:: ZF_FUND1 semantic presentation
definition
let A be
set ;
let B be
set ;
func c1 (#) c2 -> set means :
Def1:
:: ZF_FUND1:def 1
for
p being
set holds
(
p in it iff ex
q,
r,
s being
set st
(
p = [q,s] &
[q,r] in A &
[r,s] in B ) );
existence
ex b1 being set st
for p being set holds
( p in b1 iff ex q, r, s being set st
( p = [q,s] & [q,r] in A & [r,s] in B ) )
uniqueness
for b1, b2 being set st ( for p being set holds
( p in b1 iff ex q, r, s being set st
( p = [q,s] & [q,r] in A & [r,s] in B ) ) ) & ( for p being set holds
( p in b2 iff ex q, r, s being set st
( p = [q,s] & [q,r] in A & [r,s] in B ) ) ) holds
b1 = b2
end;
:: deftheorem Def1 defines (#) ZF_FUND1:def 1 :
for
A,
B,
b3 being
set holds
(
b3 = A (#) B iff for
p being
set holds
(
p in b3 iff ex
q,
r,
s being
set st
(
p = [q,s] &
[q,r] in A &
[r,s] in B ) ) );
:: deftheorem Def2 defines decode ZF_FUND1:def 2 :
:: deftheorem Def3 defines x". ZF_FUND1:def 3 :
Lemma78:
( dom decode = omega & rng decode = VAR & decode is one-to-one & decode " is one-to-one & dom (decode " ) = VAR & rng (decode " ) = omega )
:: deftheorem Def4 defines code ZF_FUND1:def 4 :
definition
let H be
ZF-formula;
let E be non
empty set ;
func Diagram c1,
c2 -> set means :
Def5:
:: ZF_FUND1:def 5
for
p being
set holds
(
p in it iff ex
f being
Function of
VAR ,
E st
(
p = (f * decode ) | (code (Free H)) &
f in St H,
E ) );
existence
ex b1 being set st
for p being set holds
( p in b1 iff ex f being Function of VAR ,E st
( p = (f * decode ) | (code (Free H)) & f in St H,E ) )
uniqueness
for b1, b2 being set st ( for p being set holds
( p in b1 iff ex f being Function of VAR ,E st
( p = (f * decode ) | (code (Free H)) & f in St H,E ) ) ) & ( for p being set holds
( p in b2 iff ex f being Function of VAR ,E st
( p = (f * decode ) | (code (Free H)) & f in St H,E ) ) ) holds
b1 = b2
end;
:: deftheorem Def5 defines Diagram ZF_FUND1:def 5 :
definition
let V be
Universe;
let X be
Subclass of
V;
attr a2 is
closed_wrt_A1 means :
Def6:
:: ZF_FUND1:def 6
for
a being
Element of
V st
a in X holds
{ {[(0-element_of V),x],[(1-element_of V),y]} where x is Element of V, y is Element of V : ( x in y & x in a & y in a ) } in X;
attr a2 is
closed_wrt_A2 means :
Def7:
:: ZF_FUND1:def 7
for
a,
b being
Element of
V st
a in X &
b in X holds
{a,b} in X;
attr a2 is
closed_wrt_A3 means :
Def8:
:: ZF_FUND1:def 8
for
a being
Element of
V st
a in X holds
union a in X;
attr a2 is
closed_wrt_A4 means :
Def9:
:: ZF_FUND1:def 9
for
a,
b being
Element of
V st
a in X &
b in X holds
{ {[x,y]} where x is Element of V, y is Element of V : ( x in a & y in b ) } in X;
attr a2 is
closed_wrt_A5 means :
Def10:
:: ZF_FUND1:def 10
for
a,
b being
Element of
V st
a in X &
b in X holds
{ (x \/ y) where x is Element of V, y is Element of V : ( x in a & y in b ) } in X;
attr a2 is
closed_wrt_A6 means :
Def11:
:: ZF_FUND1:def 11
for
a,
b being
Element of
V st
a in X &
b in X holds
{ (x \ y) where x is Element of V, y is Element of V : ( x in a & y in b ) } in X;
attr a2 is
closed_wrt_A7 means :
Def12:
:: ZF_FUND1:def 12
for
a,
b being
Element of
V st
a in X &
b in X holds
{ (x (#) y) where x is Element of V, y is Element of V : ( x in a & y in b ) } in X;
end;
:: deftheorem Def6 defines closed_wrt_A1 ZF_FUND1:def 6 :
:: deftheorem Def7 defines closed_wrt_A2 ZF_FUND1:def 7 :
:: deftheorem Def8 defines closed_wrt_A3 ZF_FUND1:def 8 :
:: deftheorem Def9 defines closed_wrt_A4 ZF_FUND1:def 9 :
:: deftheorem Def10 defines closed_wrt_A5 ZF_FUND1:def 10 :
:: deftheorem Def11 defines closed_wrt_A6 ZF_FUND1:def 11 :
:: deftheorem Def12 defines closed_wrt_A7 ZF_FUND1:def 12 :
:: deftheorem Def13 defines closed_wrt_A1-A7 ZF_FUND1:def 13 :
Lemma90:
for A being Element of omega holds A = x". (x. (card A))
Lemma91:
for fs being finite Subset of omega
for E being non empty set
for f being Function of VAR ,E holds
( dom ((f * decode ) | fs) = fs & rng ((f * decode ) | fs) c= E & (f * decode ) | fs in Funcs fs,E & dom (f * decode ) = omega )
Lemma92:
for E being non empty set
for f being Function of VAR ,E
for v1 being Element of VAR holds
( decode . (x". v1) = v1 & (decode " ) . v1 = x". v1 & (f * decode ) . (x". v1) = f . v1 )
Lemma93:
for p being set
for A being finite Subset of VAR holds
( p in code A iff ex v1 being Element of VAR st
( v1 in A & p = x". v1 ) )
Lemma94:
for v1 being Element of VAR holds code {v1} = {(x". v1)}
Lemma95:
for v1, v2 being Element of VAR holds code {v1,v2} = {(x". v1),(x". v2)}
Lemma96:
for A being finite Subset of VAR holds A, code A are_equipotent
Lemma97:
for A, B being finite Subset of VAR holds
( code (A \/ B) = (code A) \/ (code B) & code (A \ B) = (code A) \ (code B) )
by RELAT_1:153, , FUNCT_1:123;
Lemma98:
for E being non empty set
for f being Function of VAR ,E
for v1 being Element of VAR
for H being ZF-formula st v1 in Free H holds
((f * decode ) | (code (Free H))) . (x". v1) = f . v1
Lemma99:
for E being non empty set
for H being ZF-formula
for f, g being Function of VAR ,E st (f * decode ) | (code (Free H)) = (g * decode ) | (code (Free H)) & f in St H,E holds
g in St H,E
Lemma100:
for p being set
for fs being finite Subset of omega
for E being non empty set st p in Funcs fs,E holds
ex f being Function of VAR ,E st p = (f * decode ) | fs
theorem Th1: :: ZF_FUND1:1
theorem Th2: :: ZF_FUND1:2
theorem Th3: :: ZF_FUND1:3
theorem Th4: :: ZF_FUND1:4
theorem Th5: :: ZF_FUND1:5
theorem Th6: :: ZF_FUND1:6
theorem Th7: :: ZF_FUND1:7
theorem Th8: :: ZF_FUND1:8
theorem Th9: :: ZF_FUND1:9
theorem Th10: :: ZF_FUND1:10
Lemma130:
for V being Universe
for X being Subclass of V
for n being Element of omega st X is closed_wrt_A1-A7 holds
n in X
Lemma131:
for V being Universe
for X being Subclass of V st X is closed_wrt_A1-A7 holds
( 0-element_of V in X & 1-element_of V in X )
theorem Th11: :: ZF_FUND1:11
theorem Th12: :: ZF_FUND1:12
theorem Th13: :: ZF_FUND1:13
theorem Th14: :: ZF_FUND1:14
theorem Th15: :: ZF_FUND1:15
theorem Th16: :: ZF_FUND1:16
Lemma165:
for o, p, q being set holds {[o,p],[p,p]} (#) {[p,q]} = {[o,q],[p,q]}
theorem Th17: :: ZF_FUND1:17
Lemma169:
for p, r, o, q, s, t being set st p <> r holds
{[o,p],[q,r]} (#) {[p,s],[r,t]} = {[o,s],[q,t]}
Lemma170:
for o, q being set
for g being Function holds
( dom g = {o,q} iff g = {[o,(g . o)],[q,(g . q)]} )
theorem Th18: :: ZF_FUND1:18
theorem Th19: :: ZF_FUND1:19
theorem Th20: :: ZF_FUND1:20
theorem Th21: :: ZF_FUND1:21
theorem Th22: :: ZF_FUND1:22
theorem Th23: :: ZF_FUND1:23
theorem Th24: :: ZF_FUND1:24
theorem Th25: :: ZF_FUND1:25
for
p,
r,
o,
q,
s,
t being
set st
p <> r holds
{[o,p],[q,r]} (#) {[p,s],[r,t]} = {[o,s],[q,t]} by ;
theorem Th26: :: ZF_FUND1:26
canceled;
theorem Th27: :: ZF_FUND1:27
theorem Th28: :: ZF_FUND1:28
theorem Th29: :: ZF_FUND1:29
theorem Th30: :: ZF_FUND1:30
theorem Th31: :: ZF_FUND1:31
theorem Th32: :: ZF_FUND1:32
theorem Th33: :: ZF_FUND1:33
theorem Th34: :: ZF_FUND1:34
theorem Th35: :: ZF_FUND1:35
theorem Th36: :: ZF_FUND1:36
theorem Th37: :: ZF_FUND1:37
theorem Th38: :: ZF_FUND1:38