Theorem fvmpt3 5721

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 2298	. . 3 |- ( x = A -> ( B e. V <-> C e. V ) )
3		fvmpt3.c	. . 3 |- ( x e. D -> B e. V )
4	2, 3	vtoclga 2868	. 2 |- ( A e. D -> C e. V )
5		fvmpt3.b	. . 3 |- F = ( x e. D |-> B )
6	1, 5	fvmptg 5718	. 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 1395   e. wcel 2200   |-> cmpt 4148   ` cfv 5324

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332

This theorem is referenced by:  fvmpt3i  5722  frec2uzsucd  10653