Theorem fvopab3ig 5708

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 2292	. . . . . . . 8 |- ( x = A -> ( x e. C <-> A e. C ) )
2		fvopab3ig.1	. . . . . . . 8 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	anbi12d 473	. . . . . . 7 |- ( x = A -> ( ( x e. C /\ ph ) <-> ( A e. C /\ ps ) ) )
4		fvopab3ig.2	. . . . . . . 8 |- ( y = B -> ( ps <-> ch ) )
5	4	anbi2d 464	. . . . . . 7 |- ( y = B -> ( ( A e. C /\ ps ) <-> ( A e. C /\ ch ) ) )
6	3, 5	opelopabg 4356	. . . . . 6 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } <-> ( A e. C /\ ch ) ) )
7	6	biimpar 297	. . . . 5 |- ( ( ( A e. C /\ B e. D ) /\ ( A e. C /\ ch ) ) -> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } )
8	7	exp43 372	. . . 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 124	. 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 5628	. . 3 |- ( F ` A ) = ( { <. x , y >. | ( x e. C /\ ph ) } ` A )
13		funopab 5353	. . . . 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 2153	. . . . . 6 |- ( E* y ( x e. C /\ ph ) <-> ( x e. C -> E* y ph ) )
16	14, 15	mpbir 146	. . . . 5 |- E* y ( x e. C /\ ph )
17	13, 16	mpgbir 1499	. . . 4 |- Fun { <. x , y >. | ( x e. C /\ ph ) }
18		funopfv 5671	. . . 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	eqtrid 2274	. 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 104   <-> wb 105   = wceq 1395   E*wmo 2078   e. wcel 2200   <.cop 3669   {copab 4144   Fun wfun 5312   ` cfv 5318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326

This theorem is referenced by:  fvmptg  5710  fvopab6  5731  ov6g  6143