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Theorem domrancur1c 25202
Description: The currying of a mapping  F whose domain is  ( A  X.  B ) is a mapping whose domain is  A and the range, the class of all the functions from  B to  ran  F. (Contributed by FL, 17-May-2010.)
Assertion
Ref Expression
domrancur1c  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  -> 
( cur1 `  F ) : A --> { f  |  f : B --> ran  F } )
Distinct variable groups:    B, f    f, F
Allowed substitution hints:    A( f)    C( f)    D( f)

Proof of Theorem domrancur1c
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . . . . . 9  |-  x  e. 
_V
2 2ndconst 6208 . . . . . . . . 9  |-  ( x  e.  _V  ->  ( 2nd  |`  ( { x }  X.  _V ) ) : ( { x }  X.  _V ) -1-1-onto-> _V )
31, 2ax-mp 8 . . . . . . . 8  |-  ( 2nd  |`  ( { x }  X.  _V ) ) : ( { x }  X.  _V ) -1-1-onto-> _V
4 f1ocnv 5485 . . . . . . . 8  |-  ( ( 2nd  |`  ( {
x }  X.  _V ) ) : ( { x }  X.  _V ) -1-1-onto-> _V  ->  `' ( 2nd  |`  ( { x }  X.  _V ) ) : _V -1-1-onto-> ( { x }  X.  _V ) )
5 f1ofun 5474 . . . . . . . . 9  |-  ( `' ( 2nd  |`  ( { x }  X.  _V ) ) : _V -1-1-onto-> ( { x }  X.  _V )  ->  Fun  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )
6 fnfun 5341 . . . . . . . . . . . 12  |-  ( F  Fn  ( A  X.  B )  ->  Fun  F )
7 funco 5292 . . . . . . . . . . . . 13  |-  ( ( Fun  F  /\  Fun  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  ->  Fun  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) )
87ex 423 . . . . . . . . . . . 12  |-  ( Fun 
F  ->  ( Fun  `' ( 2nd  |`  ( { x }  X.  _V ) )  ->  Fun  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) ) )
96, 8syl 15 . . . . . . . . . . 11  |-  ( F  Fn  ( A  X.  B )  ->  ( Fun  `' ( 2nd  |`  ( { x }  X.  _V ) )  ->  Fun  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) ) )
109adantl 452 . . . . . . . . . 10  |-  ( ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) )  ->  ( Fun  `' ( 2nd  |`  ( { x }  X.  _V ) )  ->  Fun  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) ) )
1110ad2antlr 707 . . . . . . . . 9  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ( Fun  `' ( 2nd  |`  ( {
x }  X.  _V ) )  ->  Fun  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) ) )
125, 11syl5com 26 . . . . . . . 8  |-  ( `' ( 2nd  |`  ( { x }  X.  _V ) ) : _V -1-1-onto-> ( { x }  X.  _V )  ->  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  Fun  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) ) )
133, 4, 12mp2b 9 . . . . . . 7  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  Fun  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) )
14 dmco 5181 . . . . . . . 8  |-  dom  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  =  ( `' `' ( 2nd  |`  ( { x }  X.  _V ) ) " dom  F )
15 cnvcnvres 5136 . . . . . . . . . 10  |-  `' `' ( 2nd  |`  ( {
x }  X.  _V ) )  =  ( `' `' 2nd  |`  ( {
x }  X.  _V ) )
1615imaeq1i 5009 . . . . . . . . 9  |-  ( `' `' ( 2nd  |`  ( { x }  X.  _V ) ) " dom  F )  =  ( ( `' `' 2nd  |`  ( {
x }  X.  _V ) ) " dom  F )
17 df-ima 4702 . . . . . . . . . 10  |-  ( ( `' `' 2nd  |`  ( {
x }  X.  _V ) ) " dom  F )  =  ran  (
( `' `' 2nd  |`  ( { x }  X.  _V ) )  |`  dom  F )
18 resres 4968 . . . . . . . . . . . . 13  |-  ( ( `' `' 2nd  |`  ( {
x }  X.  _V ) )  |`  dom  F
)  =  ( `' `' 2nd  |`  ( ( { x }  X.  _V )  i^i  dom  F
) )
1918a1i 10 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ( ( `' `' 2nd  |`  ( { x }  X.  _V ) )  |`  dom  F )  =  ( `' `' 2nd  |`  ( ( { x }  X.  _V )  i^i 
dom  F ) ) )
2019rneqd 4906 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ran  ( ( `' `' 2nd  |`  ( {
x }  X.  _V ) )  |`  dom  F
)  =  ran  ( `' `' 2nd  |`  ( ( { x }  X.  _V )  i^i  dom  F
) ) )
21 rescnvcnv 5135 . . . . . . . . . . . . . 14  |-  ( `' `' 2nd  |`  ( ( { x }  X.  _V )  i^i  dom  F
) )  =  ( 2nd  |`  ( ( { x }  X.  _V )  i^i  dom  F
) )
2221a1i 10 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ( `' `' 2nd  |`  ( ( { x }  X.  _V )  i^i 
dom  F ) )  =  ( 2nd  |`  (
( { x }  X.  _V )  i^i  dom  F ) ) )
2322rneqd 4906 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ran  ( `' `' 2nd  |`  ( ( { x }  X.  _V )  i^i  dom  F )
)  =  ran  ( 2nd  |`  ( ( { x }  X.  _V )  i^i  dom  F )
) )
24 fndm 5343 . . . . . . . . . . . . . . . 16  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
25 simpr 447 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  A  /\  ( A  e.  C  /\  B  e.  D
) )  /\  dom  F  =  ( A  X.  B ) )  ->  dom  F  =  ( A  X.  B ) )
2625ineq2d 3370 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( x  e.  A  /\  ( A  e.  C  /\  B  e.  D
) )  /\  dom  F  =  ( A  X.  B ) )  -> 
( ( { x }  X.  _V )  i^i 
dom  F )  =  ( ( { x }  X.  _V )  i^i  ( A  X.  B
) ) )
2726reseq2d 4955 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( x  e.  A  /\  ( A  e.  C  /\  B  e.  D
) )  /\  dom  F  =  ( A  X.  B ) )  -> 
( 2nd  |`  ( ( { x }  X.  _V )  i^i  dom  F
) )  =  ( 2nd  |`  ( ( { x }  X.  _V )  i^i  ( A  X.  B ) ) ) )
2827rneqd 4906 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  A  /\  ( A  e.  C  /\  B  e.  D
) )  /\  dom  F  =  ( A  X.  B ) )  ->  ran  ( 2nd  |`  (
( { x }  X.  _V )  i^i  dom  F ) )  =  ran  ( 2nd  |`  ( ( { x }  X.  _V )  i^i  ( A  X.  B ) ) ) )
29 inxp 4818 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( { x }  X.  _V )  i^i  ( A  X.  B ) )  =  ( ( { x }  i^i  A
)  X.  ( _V 
i^i  B ) )
3029reseq2i 4952 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 2nd  |`  ( ( { x }  X.  _V )  i^i  ( A  X.  B
) ) )  =  ( 2nd  |`  (
( { x }  i^i  A )  X.  ( _V  i^i  B ) ) )
3130rneqi 4905 . . . . . . . . . . . . . . . . . . . 20  |-  ran  ( 2nd  |`  ( ( { x }  X.  _V )  i^i  ( A  X.  B ) ) )  =  ran  ( 2nd  |`  ( ( { x }  i^i  A )  X.  ( _V  i^i  B
) ) )
32 snssi 3759 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( x  e.  A  ->  { x }  C_  A )
33 df-ss 3166 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( { x }  C_  A  <->  ( { x }  i^i  A )  =  { x } )
3432, 33sylib 188 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( x  e.  A  ->  ( { x }  i^i  A )  =  { x } )
35 incom 3361 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( _V 
i^i  B )  =  ( B  i^i  _V )
36 inv1 3481 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( B  i^i  _V )  =  B
3735, 36eqtri 2303 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( _V 
i^i  B )  =  B
38 xpeq12 4708 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( { x }  i^i  A )  =  {
x }  /\  ( _V  i^i  B )  =  B )  ->  (
( { x }  i^i  A )  X.  ( _V  i^i  B ) )  =  ( { x }  X.  B ) )
3934, 37, 38sylancl 643 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  e.  A  ->  (
( { x }  i^i  A )  X.  ( _V  i^i  B ) )  =  ( { x }  X.  B ) )
4039reseq2d 4955 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  e.  A  ->  ( 2nd  |`  ( ( { x }  i^i  A
)  X.  ( _V 
i^i  B ) ) )  =  ( 2nd  |`  ( { x }  X.  B ) ) )
4140rneqd 4906 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  e.  A  ->  ran  ( 2nd  |`  ( ( { x }  i^i  A )  X.  ( _V 
i^i  B ) ) )  =  ran  ( 2nd  |`  ( { x }  X.  B ) ) )
4241ad2antrr 706 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( x  e.  A  /\  ( A  e.  C  /\  B  e.  D
) )  /\  dom  F  =  ( A  X.  B ) )  ->  ran  ( 2nd  |`  (
( { x }  i^i  A )  X.  ( _V  i^i  B ) ) )  =  ran  ( 2nd  |`  ( { x }  X.  B ) ) )
43 2ndconst 6208 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  e.  _V  ->  ( 2nd  |`  ( { x }  X.  B ) ) : ( { x }  X.  B ) -1-1-onto-> B )
44 f1ofo 5479 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 2nd  |`  ( {
x }  X.  B
) ) : ( { x }  X.  B ) -1-1-onto-> B  ->  ( 2nd  |`  ( { x }  X.  B ) ) : ( { x }  X.  B ) -onto-> B )
451, 43, 44mp2b 9 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 2nd  |`  ( { x }  X.  B ) ) : ( { x }  X.  B ) -onto-> B
46 forn 5454 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 2nd  |`  ( {
x }  X.  B
) ) : ( { x }  X.  B ) -onto-> B  ->  ran  ( 2nd  |`  ( { x }  X.  B ) )  =  B )
4745, 46mp1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( x  e.  A  /\  ( A  e.  C  /\  B  e.  D
) )  /\  dom  F  =  ( A  X.  B ) )  ->  ran  ( 2nd  |`  ( { x }  X.  B ) )  =  B )
4842, 47eqtrd 2315 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( x  e.  A  /\  ( A  e.  C  /\  B  e.  D
) )  /\  dom  F  =  ( A  X.  B ) )  ->  ran  ( 2nd  |`  (
( { x }  i^i  A )  X.  ( _V  i^i  B ) ) )  =  B )
4931, 48syl5eq 2327 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  A  /\  ( A  e.  C  /\  B  e.  D
) )  /\  dom  F  =  ( A  X.  B ) )  ->  ran  ( 2nd  |`  (
( { x }  X.  _V )  i^i  ( A  X.  B ) ) )  =  B )
5028, 49eqtrd 2315 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  A  /\  ( A  e.  C  /\  B  e.  D
) )  /\  dom  F  =  ( A  X.  B ) )  ->  ran  ( 2nd  |`  (
( { x }  X.  _V )  i^i  dom  F ) )  =  B )
5150exp31 587 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  A  ->  (
( A  e.  C  /\  B  e.  D
)  ->  ( dom  F  =  ( A  X.  B )  ->  ran  ( 2nd  |`  ( ( { x }  X.  _V )  i^i  dom  F
) )  =  B ) ) )
5251com13 74 . . . . . . . . . . . . . . . 16  |-  ( dom 
F  =  ( A  X.  B )  -> 
( ( A  e.  C  /\  B  e.  D )  ->  (
x  e.  A  ->  ran  ( 2nd  |`  (
( { x }  X.  _V )  i^i  dom  F ) )  =  B ) ) )
5324, 52syl 15 . . . . . . . . . . . . . . 15  |-  ( F  Fn  ( A  X.  B )  ->  (
( A  e.  C  /\  B  e.  D
)  ->  ( x  e.  A  ->  ran  ( 2nd  |`  ( ( { x }  X.  _V )  i^i  dom  F )
)  =  B ) ) )
5453adantl 452 . . . . . . . . . . . . . 14  |-  ( ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) )  ->  (
( A  e.  C  /\  B  e.  D
)  ->  ( x  e.  A  ->  ran  ( 2nd  |`  ( ( { x }  X.  _V )  i^i  dom  F )
)  =  B ) ) )
5554impcom 419 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  -> 
( x  e.  A  ->  ran  ( 2nd  |`  (
( { x }  X.  _V )  i^i  dom  F ) )  =  B ) )
5655imp 418 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ran  ( 2nd  |`  (
( { x }  X.  _V )  i^i  dom  F ) )  =  B )
5723, 56eqtrd 2315 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ran  ( `' `' 2nd  |`  ( ( { x }  X.  _V )  i^i  dom  F )
)  =  B )
5820, 57eqtrd 2315 . . . . . . . . . 10  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ran  ( ( `' `' 2nd  |`  ( {
x }  X.  _V ) )  |`  dom  F
)  =  B )
5917, 58syl5eq 2327 . . . . . . . . 9  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ( ( `' `' 2nd  |`  ( { x }  X.  _V ) )
" dom  F )  =  B )
6016, 59syl5eq 2327 . . . . . . . 8  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ( `' `' ( 2nd  |`  ( {
x }  X.  _V ) ) " dom  F )  =  B )
6114, 60syl5eq 2327 . . . . . . 7  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  dom  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  =  B )
62 df-fn 5258 . . . . . . 7  |-  ( ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  Fn  B  <->  ( Fun  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  /\  dom  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  =  B ) )
6313, 61, 62sylanbrc 645 . . . . . 6  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  Fn  B )
64 dffn3 5396 . . . . . 6  |-  ( ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  Fn  B  <->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) : B --> ran  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) )
6563, 64sylib 188 . . . . 5  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) : B --> ran  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) )
66 rncoss 4945 . . . . 5  |-  ran  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  C_  ran  F
67 fss 5397 . . . . 5  |-  ( ( ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) : B --> ran  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  /\  ran  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  C_  ran  F )  ->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) : B --> ran  F
)
6865, 66, 67sylancl 643 . . . 4  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) : B --> ran  F )
69 xpexg 4800 . . . . . . . . . 10  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
70 fnex 5741 . . . . . . . . . . 11  |-  ( ( F  Fn  ( A  X.  B )  /\  ( A  X.  B
)  e.  _V )  ->  F  e.  _V )
7170ex 423 . . . . . . . . . 10  |-  ( F  Fn  ( A  X.  B )  ->  (
( A  X.  B
)  e.  _V  ->  F  e.  _V ) )
7269, 71syl5 28 . . . . . . . . 9  |-  ( F  Fn  ( A  X.  B )  ->  (
( A  e.  C  /\  B  e.  D
)  ->  F  e.  _V ) )
7372adantl 452 . . . . . . . 8  |-  ( ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) )  ->  (
( A  e.  C  /\  B  e.  D
)  ->  F  e.  _V ) )
7473impcom 419 . . . . . . 7  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  ->  F  e.  _V )
7574adantr 451 . . . . . 6  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  F  e.  _V )
76 fo2nd 6140 . . . . . . . 8  |-  2nd : _V -onto-> _V
77 fofun 5452 . . . . . . . 8  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
78 funres 5293 . . . . . . . 8  |-  ( Fun 
2nd  ->  Fun  ( 2nd  |`  ( { x }  X.  _V ) ) )
7976, 77, 78mp2b 9 . . . . . . 7  |-  Fun  ( 2nd  |`  ( { x }  X.  _V ) )
80 funcnvcnv 5308 . . . . . . 7  |-  ( Fun  ( 2nd  |`  ( { x }  X.  _V ) )  ->  Fun  `' `' ( 2nd  |`  ( { x }  X.  _V ) ) )
8179, 80mp1i 11 . . . . . 6  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  Fun  `' `' ( 2nd  |`  ( {
x }  X.  _V ) ) )
82 cofunex2g 5740 . . . . . 6  |-  ( ( F  e.  _V  /\  Fun  `' `' ( 2nd  |`  ( { x }  X.  _V ) ) )  -> 
( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  e. 
_V )
8375, 81, 82syl2anc 642 . . . . 5  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  e. 
_V )
84 feq1 5375 . . . . . 6  |-  ( f  =  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  -> 
( f : B --> ran  F  <->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) : B --> ran  F )
)
8584elabg 2915 . . . . 5  |-  ( ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  e. 
_V  ->  ( ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  e.  { f  |  f : B --> ran  F } 
<->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) : B --> ran  F )
)
8683, 85syl 15 . . . 4  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ( ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  e. 
{ f  |  f : B --> ran  F } 
<->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) : B --> ran  F )
)
8768, 86mpbird 223 . . 3  |-  ( ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  /\  x  e.  A )  ->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  e. 
{ f  |  f : B --> ran  F } )
8887ralrimiva 2626 . 2  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  ->  A. x  e.  A  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  e. 
{ f  |  f : B --> ran  F } )
89 pm3.22 436 . . . . 5  |-  ( ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) )  ->  ( F  Fn  ( A  X.  B )  /\  B  =/=  (/) ) )
9089adantl 452 . . . 4  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  -> 
( F  Fn  ( A  X.  B )  /\  B  =/=  (/) ) )
91 simpl 443 . . . 4  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  -> 
( A  e.  C  /\  B  e.  D
) )
92 cur1vald 25199 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  B  =/=  (/) )  /\  ( A  e.  C  /\  B  e.  D )
)  ->  ( cur1 `  F )  =  ( x  e.  A  |->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) ) )
9390, 91, 92syl2anc 642 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  -> 
( cur1 `  F )  =  ( x  e.  A  |->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) ) )
94 fopab2g 25145 . . 3  |-  ( (
cur1 `  F )  =  ( x  e.  A  |->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) )  ->  ( A. x  e.  A  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  e. 
{ f  |  f : B --> ran  F } 
<->  ( cur1 `  F
) : A --> { f  |  f : B --> ran  F } ) )
9593, 94syl 15 . 2  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  -> 
( A. x  e.  A  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) )  e. 
{ f  |  f : B --> ran  F } 
<->  ( cur1 `  F
) : A --> { f  |  f : B --> ran  F } ) )
9688, 95mpbid 201 1  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B
) ) )  -> 
( cur1 `  F ) : A --> { f  |  f : B --> ran  F } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    o. ccom 4693   Fun wfun 5249    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255   2ndc2nd 6121   cur1ccur1 25194
This theorem is referenced by:  valcurfn  25203  curgrpact  25372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6122  df-2nd 6123  df-cur1 25196
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