Theorem fvopab6 5774

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 2825	. . 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 2244	. . . . 5 |- ( x = A -> ( y = B <-> y = C ) )
5	2, 4	anbi12d 473	. . . 4 |- ( x = A -> ( ( ph /\ y = B ) <-> ( ps /\ y = C ) ) )
6		iba 300	. . . . 5 |- ( y = C -> ( ps <-> ( ps /\ y = C ) ) )
7	6	bicomd 141	. . . 4 |- ( y = C -> ( ( ps /\ y = C ) <-> ps ) )
8		moeq 2992	. . . . . 6 |- E* y y = B
9	8	moani 2151	. . . . 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 2816	. . . . . . 7 |- x e. _V
13	12	biantrur 303	. . . . . 6 |- ( ( ph /\ y = B ) <-> ( x e. _V /\ ( ph /\ y = B ) ) )
14	13	opabbii 4177	. . . . 5 |- { <. x , y >. | ( ph /\ y = B ) } = { <. x , y >. | ( x e. _V /\ ( ph /\ y = B ) ) }
15	11, 14	eqtri 2253	. . . 4 |- F = { <. x , y >. | ( x e. _V /\ ( ph /\ y = B ) ) }
16	5, 7, 10, 15	fvopab3ig 5751	. . 3 |- ( ( A e. _V /\ C e. R ) -> ( ps -> ( F ` A ) = C ) )
17	1, 16	sylan 283	. 2 |- ( ( A e. D /\ C e. R ) -> ( ps -> ( F ` A ) = C ) )
18	17	3impia 1227	1 |- ( ( A e. D /\ C e. R /\ ps ) -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   E*wmo 2081   e. wcel 2203   _Vcvv 2813   {copab 4170   ` cfv 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360

This theorem is referenced by: (None)