Theorem fvopab6 5285

Description: Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)

Hypotheses

Ref	Expression
fvopab6.1	|- F = { <. x , y >. | ( ph /\ y = B ) }
fvopab6.2	|- ( x = A -> ( ph <-> ps ) )
fvopab6.3	|- ( x = A -> B = C )

Assertion

Ref	Expression
fvopab6	|- ( ( A e. D /\ C e. R /\ ps ) -> ( F ` A ) = C )

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

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

Proof of Theorem fvopab6

Step	Hyp	Ref	Expression
1		elex 2610	. . 3 |- ( A e. D -> A e. _V )
2		fvopab6.2	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
3		fvopab6.3	. . . . . 6 |- ( x = A -> B = C )
4	3	eqeq2d 2092	. . . . 5 |- ( x = A -> ( y = B <-> y = C ) )
5	2, 4	anbi12d 456	. . . 4 |- ( x = A -> ( ( ph /\ y = B ) <-> ( ps /\ y = C ) ) )
6		iba 294	. . . . 5 |- ( y = C -> ( ps <-> ( ps /\ y = C ) ) )
7	6	bicomd 139	. . . 4 |- ( y = C -> ( ( ps /\ y = C ) <-> ps ) )
8		moeq 2767	. . . . . 6 |- E* y y = B
9	8	moani 2011	. . . . 5 |- E* y ( ph /\ y = B )
10	9	a1i 9	. . . 4 |- ( x e. _V -> E* y ( ph /\ y = B ) )
11		fvopab6.1	. . . . 5 |- F = { <. x , y >. | ( ph /\ y = B ) }
12		vex 2604	. . . . . . 7 |- x e. _V
13	12	biantrur 297	. . . . . 6 |- ( ( ph /\ y = B ) <-> ( x e. _V /\ ( ph /\ y = B ) ) )
14	13	opabbii 3845	. . . . 5 |- { <. x , y >. | ( ph /\ y = B ) } = { <. x , y >. | ( x e. _V /\ ( ph /\ y = B ) ) }
15	11, 14	eqtri 2101	. . . 4 |- F = { <. x , y >. | ( x e. _V /\ ( ph /\ y = B ) ) }
16	5, 7, 10, 15	fvopab3ig 5267	. . 3 |- ( ( A e. _V /\ C e. R ) -> ( ps -> ( F ` A ) = C ) )
17	1, 16	sylan 277	. 2 |- ( ( A e. D /\ C e. R ) -> ( ps -> ( F ` A ) = C ) )
18	17	3impia 1135	1 |- ( ( A e. D /\ C e. R /\ ps ) -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   E*wmo 1942   _Vcvv 2601   {copab 3838   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930

This theorem is referenced by: (None)