Theorem fvmpt3 5587

Description: Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Hypotheses

Ref	Expression
fvmpt3.a	|- ( x = A -> B = C )
fvmpt3.b	|- F = ( x e. D |-> B )
fvmpt3.c	|- ( x e. D -> B e. V )

Assertion

Ref	Expression
fvmpt3	|- ( A e. D -> ( F ` A ) = C )

Distinct variable groups:   x, A   x, C   x, D   x, V

Allowed substitution hints:   B( x)   F( x)

Proof of Theorem fvmpt3

Step	Hyp	Ref	Expression
1		fvmpt3.a	. . . 4 |- ( x = A -> B = C )
2	1	eleq1d 2244	. . 3 |- ( x = A -> ( B e. V <-> C e. V ) )
3		fvmpt3.c	. . 3 |- ( x e. D -> B e. V )
4	2, 3	vtoclga 2801	. 2 |- ( A e. D -> C e. V )
5		fvmpt3.b	. . 3 |- F = ( x e. D |-> B )
6	1, 5	fvmptg 5584	. 2 |- ( ( A e. D /\ C e. V ) -> ( F ` A ) = C )
7	4, 6	mpdan 421	1 |- ( A e. D -> ( F ` A ) = C )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2146   |-> cmpt 4059   ` cfv 5208

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216

This theorem is referenced by:  fvmpt3i  5588  frec2uzsucd  10369