Theorem fvopab6 5658

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 2774	. . 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 2208	. . . . 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 2939	. . . . . 6 |- E* y y = B
9	8	moani 2115	. . . . 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 2766	. . . . . . 7 |- x e. _V
13	12	biantrur 303	. . . . . 6 |- ( ( ph /\ y = B ) <-> ( x e. _V /\ ( ph /\ y = B ) ) )
14	13	opabbii 4100	. . . . 5 |- { <. x , y >. | ( ph /\ y = B ) } = { <. x , y >. | ( x e. _V /\ ( ph /\ y = B ) ) }
15	11, 14	eqtri 2217	. . . 4 |- F = { <. x , y >. | ( x e. _V /\ ( ph /\ y = B ) ) }
16	5, 7, 10, 15	fvopab3ig 5635	. . 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 1202	1 |- ( ( A e. D /\ C e. R /\ ps ) -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   E*wmo 2046   e. wcel 2167   _Vcvv 2763   {copab 4093   ` cfv 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266

This theorem is referenced by: (None)