Theorem fvopab3ig 5488

Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)

Hypotheses

Ref	Expression
fvopab3ig.1	|- ( x = A -> ( ph <-> ps ) )
fvopab3ig.2	|- ( y = B -> ( ps <-> ch ) )
fvopab3ig.3	|- ( x e. C -> E* y ph )
fvopab3ig.4	|- F = { <. x , y >. | ( x e. C /\ ph ) }

Assertion

Ref	Expression
fvopab3ig	|- ( ( A e. C /\ B e. D ) -> ( ch -> ( F ` A ) = B ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   ch, x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   D( x, y)   F( x, y)

Proof of Theorem fvopab3ig

Step	Hyp	Ref	Expression
1		eleq1 2200	. . . . . . . 8 |- ( x = A -> ( x e. C <-> A e. C ) )
2		fvopab3ig.1	. . . . . . . 8 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	anbi12d 464	. . . . . . 7 |- ( x = A -> ( ( x e. C /\ ph ) <-> ( A e. C /\ ps ) ) )
4		fvopab3ig.2	. . . . . . . 8 |- ( y = B -> ( ps <-> ch ) )
5	4	anbi2d 459	. . . . . . 7 |- ( y = B -> ( ( A e. C /\ ps ) <-> ( A e. C /\ ch ) ) )
6	3, 5	opelopabg 4185	. . . . . 6 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } <-> ( A e. C /\ ch ) ) )
7	6	biimpar 295	. . . . 5 |- ( ( ( A e. C /\ B e. D ) /\ ( A e. C /\ ch ) ) -> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } )
8	7	exp43 369	. . . 4 |- ( A e. C -> ( B e. D -> ( A e. C -> ( ch -> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } ) ) ) )
9	8	pm2.43a 51	. . 3 |- ( A e. C -> ( B e. D -> ( ch -> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } ) ) )
10	9	imp 123	. 2 |- ( ( A e. C /\ B e. D ) -> ( ch -> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } ) )
11		fvopab3ig.4	. . . 4 |- F = { <. x , y >. | ( x e. C /\ ph ) }
12	11	fveq1i 5415	. . 3 |- ( F ` A ) = ( { <. x , y >. | ( x e. C /\ ph ) } ` A )
13		funopab 5153	. . . . 5 |- ( Fun { <. x , y >. | ( x e. C /\ ph ) } <-> A. x E* y ( x e. C /\ ph ) )
14		fvopab3ig.3	. . . . . 6 |- ( x e. C -> E* y ph )
15		moanimv 2072	. . . . . 6 |- ( E* y ( x e. C /\ ph ) <-> ( x e. C -> E* y ph ) )
16	14, 15	mpbir 145	. . . . 5 |- E* y ( x e. C /\ ph )
17	13, 16	mpgbir 1429	. . . 4 |- Fun { <. x , y >. | ( x e. C /\ ph ) }
18		funopfv 5454	. . . 4 |- ( Fun { <. x , y >. | ( x e. C /\ ph ) } -> ( <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } -> ( { <. x , y >. | ( x e. C /\ ph ) } ` A ) = B ) )
19	17, 18	ax-mp 5	. . 3 |- ( <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } -> ( { <. x , y >. | ( x e. C /\ ph ) } ` A ) = B )
20	12, 19	syl5eq 2182	. 2 |- ( <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } -> ( F ` A ) = B )
21	10, 20	syl6 33	1 |- ( ( A e. C /\ B e. D ) -> ( ch -> ( F ` A ) = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   E*wmo 1998   <.cop 3525   {copab 3983   Fun wfun 5112   ` cfv 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126

This theorem is referenced by:  fvmptg  5490  fvopab6  5510  ov6g  5901