Theorem fvopab6 5592

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 2741	. . 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 2182	. . . . 5 |- ( x = A -> ( y = B <-> y = C ) )
5	2, 4	anbi12d 470	. . . 4 |- ( x = A -> ( ( ph /\ y = B ) <-> ( ps /\ y = C ) ) )
6		iba 298	. . . . 5 |- ( y = C -> ( ps <-> ( ps /\ y = C ) ) )
7	6	bicomd 140	. . . 4 |- ( y = C -> ( ( ps /\ y = C ) <-> ps ) )
8		moeq 2905	. . . . . 6 |- E* y y = B
9	8	moani 2089	. . . . 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 2733	. . . . . . 7 |- x e. _V
13	12	biantrur 301	. . . . . 6 |- ( ( ph /\ y = B ) <-> ( x e. _V /\ ( ph /\ y = B ) ) )
14	13	opabbii 4056	. . . . 5 |- { <. x , y >. | ( ph /\ y = B ) } = { <. x , y >. | ( x e. _V /\ ( ph /\ y = B ) ) }
15	11, 14	eqtri 2191	. . . 4 |- F = { <. x , y >. | ( x e. _V /\ ( ph /\ y = B ) ) }
16	5, 7, 10, 15	fvopab3ig 5570	. . 3 |- ( ( A e. _V /\ C e. R ) -> ( ps -> ( F ` A ) = C ) )
17	1, 16	sylan 281	. 2 |- ( ( A e. D /\ C e. R ) -> ( ps -> ( F ` A ) = C ) )
18	17	3impia 1195	1 |- ( ( A e. D /\ C e. R /\ ps ) -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   E*wmo 2020   e. wcel 2141   _Vcvv 2730   {copab 4049   ` cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206

This theorem is referenced by: (None)