:: FRAENKEL semantic presentation
scheme :: FRAENKEL:sch 2
s2{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3(
set ,
set )
-> set , P
1[
set ,
set ], P
2[
set ,
set ] } :
{ F3(b1,b2) where B is Element of F1(), B is Element of F2() : P1[b1,b2] } c= { F3(b1,b2) where B is Element of F1(), B is Element of F2() : P2[b1,b2] }
provided
E1:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds
( P
1[b
1,b
2] implies P
2[b
1,b
2] )
scheme :: FRAENKEL:sch 4
s4{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3(
set ,
set )
-> set , P
1[
set ,
set ], P
2[
set ,
set ] } :
{ F3(b1,b2) where B is Element of F1(), B is Element of F2() : P1[b1,b2] } = { F3(b1,b2) where B is Element of F1(), B is Element of F2() : P2[b1,b2] }
provided
E1:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds
( P
1[b
1,b
2] iff P
2[b
1,b
2] )
scheme :: FRAENKEL:sch 7
s7{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3(
set ,
set )
-> set , F
4(
set ,
set )
-> set , P
1[
set ,
set ] } :
{ F3(b1,b2) where B is Element of F1(), B is Element of F2() : P1[b1,b2] } = { F4(b1,b2) where B is Element of F1(), B is Element of F2() : P1[b1,b2] }
provided
E1:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds F
3(b
1,b
2)
= F
4(b
1,b
2)
scheme :: FRAENKEL:sch 8
s8{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3(
set ,
set )
-> set , P
1[
set ,
set ], P
2[
set ,
set ] } :
{ F3(b1,b2) where B is Element of F1(), B is Element of F2() : P1[b1,b2] } = { F3(b2,b1) where B is Element of F1(), B is Element of F2() : P2[b1,b2] }
provided
E1:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds
( P
1[b
1,b
2] iff P
2[b
1,b
2] )
and
E2:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds F
3(b
1,b
2)
= F
3(b
2,b
1)
theorem Th1: :: FRAENKEL:1
canceled;
theorem Th2: :: FRAENKEL:2
canceled;
theorem Th3: :: FRAENKEL:3
for b
1, b
2 being non
empty set for b
3, b
4 being
Function of b
1,b
2for b
5 being
set holds
( b
3 | b
5 = b
4 | b
5 implies for b
6 being
Element of b
1 holds
( b
6 in b
5 implies b
3 . b
6 = b
4 . b
6 ) )
theorem Th4: :: FRAENKEL:4
canceled;
theorem Th5: :: FRAENKEL:5
theorem Th6: :: FRAENKEL:6
scheme :: FRAENKEL:sch 11
s11{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3(
set ,
set )
-> set , P
1[
set ,
set ], P
2[
set ] } :
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds
( P
1[b
1,b
2] implies P
2[F
3(b
1,b
2)] )
provided
E4:
for b
1 being
set holds
( b
1 in { F3(b2,b3) where B is Element of F1(), B is Element of F2() : P1[b2,b3] } implies P
2[b
1] )
scheme :: FRAENKEL:sch 12
s12{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3(
set ,
set )
-> set , P
1[
set ,
set ], P
2[
set ] } :
for b
1 being
set holds
( b
1 in { F3(b2,b3) where B is Element of F1(), B is Element of F2() : P1[b2,b3] } implies P
2[b
1] )
provided
E4:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds
( P
1[b
1,b
2] implies P
2[F
3(b
1,b
2)] )
scheme :: FRAENKEL:sch 13
s13{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3()
-> non
empty set , F
4(
set ,
set )
-> Element of F
3(), P
1[
set ,
set ], P
2[
set ] } :
{ b1 where B is Element of F3() : ( b1 in { F4(b2,b3) where B is Element of F1(), B is Element of F2() : P1[b2,b3] } & P2[b1] ) } = { F4(b1,b2) where B is Element of F1(), B is Element of F2() : ( P1[b1,b2] & P2[F4(b1,b2)] ) }
scheme :: FRAENKEL:sch 15
s15{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3(
set ,
set )
-> set , P
1[
set ,
set ], P
2[
set ] } :
{ F3(b1,b2) where B is Element of F1(), B is Element of F2() : ( b1 in { b3 where B is Element of F1() : P2[b3] } & P1[b1,b2] ) } = { F3(b1,b2) where B is Element of F1(), B is Element of F2() : ( P2[b1] & P1[b1,b2] ) }
scheme :: FRAENKEL:sch 16
s16{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3(
set ,
set )
-> set , P
1[
set ,
set ], P
2[
set ,
set ] } :
{ F3(b1,b2) where B is Element of F1(), B is Element of F2() : P1[b1,b2] } c= { F3(b1,b2) where B is Element of F1(), B is Element of F2() : P2[b1,b2] }
provided
E4:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds
not ( P
1[b
1,b
2] & ( for b
3 being
Element of F
1() holds
not ( P
2[b
3,b
2] & F
3(b
1,b
2)
= F
3(b
3,b
2) ) ) )
scheme :: FRAENKEL:sch 19
s19{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3(
set ,
set )
-> set , F
4()
-> Element of F
2(), P
1[
set ,
set ], P
2[
set ,
set ] } :
{ F3(b1,b2) where B is Element of F1(), B is Element of F2() : P2[b1,b2] } = { F3(b1,F4()) where B is Element of F1() : P1[b1,F4()] }
provided
E4:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds
( P
2[b
1,b
2] iff ( b
2 = F
4() & P
1[b
1,b
2] ) )
scheme :: FRAENKEL:sch 20
s20{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3(
set ,
set )
-> set , F
4()
-> Element of F
2(), P
1[
set ,
set ] } :
{ F3(b1,b2) where B is Element of F1(), B is Element of F2() : ( b2 = F4() & P1[b1,b2] ) } = { F3(b1,F4()) where B is Element of F1() : P1[b1,F4()] }
:: deftheorem Def1 defines functional FRAENKEL:def 1 :
theorem Th7: :: FRAENKEL:7
canceled;
theorem Th8: :: FRAENKEL:8
:: deftheorem Def2 defines FUNCTION_DOMAIN FRAENKEL:def 2 :
theorem Th9: :: FRAENKEL:9
canceled;
theorem Th10: :: FRAENKEL:10
theorem Th11: :: FRAENKEL:11
theorem Th12: :: FRAENKEL:12
canceled;
theorem Th13: :: FRAENKEL:13
canceled;
theorem Th14: :: FRAENKEL:14
theorem Th15: :: FRAENKEL:15
scheme :: FRAENKEL:sch 23
s23{ F
1()
-> non
empty set , F
2()
-> Element of
Fin F
1(), P
1[
Element of F
1(),
Element of F
1()] } :
for b
1 being
Element of F
1() holds
not ( b
1 in F
2() & ( for b
2 being
Element of F
1() holds
not ( b
2 in F
2() & P
1[b
2,b
1] & ( for b
3 being
Element of F
1() holds
( b
3 in F
2() & P
1[b
3,b
2] implies P
1[b
2,b
3] ) ) ) ) )
provided
E10:
for b
1 being
Element of F
1() holds P
1[b
1,b
1]
and
E11:
for b
1, b
2, b
3 being
Element of F
1() holds
( P
1[b
1,b
2] & P
1[b
2,b
3] implies P
1[b
1,b
3] )
theorem Th16: :: FRAENKEL:16
theorem Th17: :: FRAENKEL:17