:: FRAENKEL semantic presentation
scheme :: FRAENKEL:sch 20
s20{
F1()
-> non
empty set ,
F2()
-> non
empty set ,
F3(
set ,
set )
-> set ,
P1[
set ,
set ],
P2[
set ,
set ] } :
{ F3(u1,v1) where u1 is Element of F1(), v1 is Element of F2() : P1[u1,v1] } c= { F3(u2,v2) where u2 is Element of F1(), v2 is Element of F2() : P2[u2,v2] }
provided
E16:
for
u being
Element of
F1()
for
v being
Element of
F2() st
P1[
u,
v] holds
P2[
u,
v]
scheme :: FRAENKEL:sch 27
s27{
F1()
-> non
empty set ,
F2()
-> non
empty set ,
F3(
set ,
set )
-> set ,
P1[
set ,
set ],
P2[
set ,
set ] } :
{ F3(u1,v1) where u1 is Element of F1(), v1 is Element of F2() : P1[u1,v1] } = { F3(u2,v2) where u2 is Element of F1(), v2 is Element of F2() : P2[u2,v2] }
provided
E16:
for
u being
Element of
F1()
for
v being
Element of
F2() holds
(
P1[
u,
v] iff
P2[
u,
v] )
scheme :: FRAENKEL:sch 33
s33{
F1()
-> non
empty set ,
F2()
-> non
empty set ,
F3(
set ,
set )
-> set ,
F4(
set ,
set )
-> set ,
P1[
set ,
set ] } :
{ F3(u1,v1) where u1 is Element of F1(), v1 is Element of F2() : P1[u1,v1] } = { F4(u2,v2) where u2 is Element of F1(), v2 is Element of F2() : P1[u2,v2] }
provided
E16:
for
u being
Element of
F1()
for
v being
Element of
F2() holds
F3(
u,
v)
= F4(
u,
v)
scheme :: FRAENKEL:sch 34
s34{
F1()
-> non
empty set ,
F2()
-> non
empty set ,
F3(
set ,
set )
-> set ,
P1[
set ,
set ],
P2[
set ,
set ] } :
{ F3(u1,v1) where u1 is Element of F1(), v1 is Element of F2() : P1[u1,v1] } = { F3(v2,u2) where u2 is Element of F1(), v2 is Element of F2() : P2[u2,v2] }
provided
E16:
for
u being
Element of
F1()
for
v being
Element of
F2() holds
(
P1[
u,
v] iff
P2[
u,
v] )
and E18:
for
u being
Element of
F1()
for
v being
Element of
F2() holds
F3(
u,
v)
= F3(
v,
u)
theorem Th1: :: FRAENKEL:1
canceled;
theorem Th2: :: FRAENKEL:2
canceled;
theorem Th3: :: FRAENKEL:3
theorem Th4: :: FRAENKEL:4
canceled;
theorem Th5: :: FRAENKEL:5
theorem Th6: :: FRAENKEL:6
scheme :: FRAENKEL:sch 48
s48{
F1()
-> non
empty set ,
F2()
-> non
empty set ,
F3(
set ,
set )
-> set ,
P1[
set ,
set ],
P2[
set ] } :
for
s being
Element of
F1()
for
t being
Element of
F2() st
P1[
s,
t] holds
P2[
F3(
s,
t)]
provided
E16:
for
st1 being
set st
st1 in { F3(s1,t1) where s1 is Element of F1(), t1 is Element of F2() : P1[s1,t1] } holds
P2[
st1]
scheme :: FRAENKEL:sch 54
s54{
F1()
-> non
empty set ,
F2()
-> non
empty set ,
F3(
set ,
set )
-> set ,
P1[
set ,
set ],
P2[
set ] } :
for
st1 being
set st
st1 in { F3(s1,t1) where s1 is Element of F1(), t1 is Element of F2() : P1[s1,t1] } holds
P2[
st1]
provided
E16:
for
s being
Element of
F1()
for
t being
Element of
F2() st
P1[
s,
t] holds
P2[
F3(
s,
t)]
scheme :: FRAENKEL:sch 55
s55{
F1()
-> non
empty set ,
F2()
-> non
empty set ,
F3()
-> non
empty set ,
F4(
set ,
set )
-> Element of
F3(),
P1[
set ,
set ],
P2[
set ] } :
{ st1 where st1 is Element of F3() : ( st1 in { F4(s1,t1) where s1 is Element of F1(), t1 is Element of F2() : P1[s1,t1] } & P2[st1] ) } = { F4(s2,t2) where s2 is Element of F1(), t2 is Element of F2() : ( P1[s2,t2] & P2[F4(s2,t2)] ) }
scheme :: FRAENKEL:sch 59
s59{
F1()
-> non
empty set ,
F2()
-> non
empty set ,
F3(
set ,
set )
-> set ,
P1[
set ,
set ],
P2[
set ] } :
{ F3(s,t) where s is Element of F1(), t is Element of F2() : ( s in { s1 where s1 is Element of F1() : P2[s1] } & P1[s,t] ) } = { F3(s2,t2) where s2 is Element of F1(), t2 is Element of F2() : ( P2[s2] & P1[s2,t2] ) }
scheme :: FRAENKEL:sch 60
s60{
F1()
-> non
empty set ,
F2()
-> non
empty set ,
F3(
set ,
set )
-> set ,
P1[
set ,
set ],
P2[
set ,
set ] } :
{ F3(s,t) where s is Element of F1(), t is Element of F2() : P1[s,t] } c= { F3(s1,t1) where s1 is Element of F1(), t1 is Element of F2() : P2[s1,t1] }
provided
E16:
for
s being
Element of
F1()
for
t being
Element of
F2() st
P1[
s,
t] holds
ex
s' being
Element of
F1() st
(
P2[
s',
t] &
F3(
s,
t)
= F3(
s',
t) )
scheme :: FRAENKEL:sch 67
s67{
F1()
-> non
empty set ,
F2()
-> non
empty set ,
F3(
set ,
set )
-> set ,
F4()
-> Element of
F2(),
P1[
set ,
set ],
P2[
set ,
set ] } :
{ F3(s,t) where s is Element of F1(), t is Element of F2() : P2[s,t] } = { F3(s',F4()) where s' is Element of F1() : P1[s',F4()] }
provided
E16:
for
s being
Element of
F1()
for
t being
Element of
F2() holds
(
P2[
s,
t] iff (
t = F4() &
P1[
s,
t] ) )
scheme :: FRAENKEL:sch 68
s68{
F1()
-> non
empty set ,
F2()
-> non
empty set ,
F3(
set ,
set )
-> set ,
F4()
-> Element of
F2(),
P1[
set ,
set ] } :
{ F3(s,t) where s is Element of F1(), t is Element of F2() : ( t = F4() & P1[s,t] ) } = { F3(s',F4()) where s' is Element of F1() : P1[s',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 89
s89{
F1()
-> non
empty set ,
F2()
-> Element of
Fin F1(),
P1[
Element of
F1(),
Element of
F1()] } :
for
x being
Element of
F1() st
x in F2() holds
ex
y being
Element of
F1() st
(
y in F2() &
P1[
y,
x] & ( for
z being
Element of
F1() st
z in F2() &
P1[
z,
y] holds
P1[
y,
z] ) )
provided
E16:
for
x being
Element of
F1() holds
P1[
x,
x]
and E18:
for
x,
y,
z being
Element of
F1() st
P1[
x,
y] &
P1[
y,
z] holds
P1[
x,
z]
theorem Th16: :: FRAENKEL:16
theorem Th17: :: FRAENKEL:17