Theorem fvopab6 5678

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 2783	. . 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 2217	. . . . 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 2948	. . . . . 6 |- E* y y = B
9	8	moani 2124	. . . . 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 2775	. . . . . . 7 |- x e. _V
13	12	biantrur 303	. . . . . 6 |- ( ( ph /\ y = B ) <-> ( x e. _V /\ ( ph /\ y = B ) ) )
14	13	opabbii 4112	. . . . 5 |- { <. x , y >. | ( ph /\ y = B ) } = { <. x , y >. | ( x e. _V /\ ( ph /\ y = B ) ) }
15	11, 14	eqtri 2226	. . . 4 |- F = { <. x , y >. | ( x e. _V /\ ( ph /\ y = B ) ) }
16	5, 7, 10, 15	fvopab3ig 5655	. . 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 1203	1 |- ( ( A e. D /\ C e. R /\ ps ) -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   E*wmo 2055   e. wcel 2176   _Vcvv 2772   {copab 4105   ` cfv 5272

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fv 5280

This theorem is referenced by: (None)