Theorem fvmpt3 5725

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 2300	. . 3 |- ( x = A -> ( B e. V <-> C e. V ) )
3		fvmpt3.c	. . 3 |- ( x e. D -> B e. V )
4	2, 3	vtoclga 2870	. 2 |- ( A e. D -> C e. V )
5		fvmpt3.b	. . 3 |- F = ( x e. D |-> B )
6	1, 5	fvmptg 5722	. 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 1397   e. wcel 2202   |-> cmpt 4150   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334

This theorem is referenced by:  fvmpt3i  5726  frec2uzsucd  10662