Theorem fvmpt3 5758

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 2303	. . 3 |- ( x = A -> ( B e. V <-> C e. V ) )
3		fvmpt3.c	. . 3 |- ( x e. D -> B e. V )
4	2, 3	vtoclga 2883	. 2 |- ( A e. D -> C e. V )
5		fvmpt3.b	. . 3 |- F = ( x e. D |-> B )
6	1, 5	fvmptg 5755	. 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 1398   e. wcel 2205   |-> cmpt 4173   ` cfv 5354

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362

This theorem is referenced by:  fvmpt3i  5759  frec2uzsucd  10767