Theorem fvmpt3 5272

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 2147	. . 3 |- ( x = A -> ( B e. V <-> C e. V ) )
3		fvmpt3.c	. . 3 |- ( x e. D -> B e. V )
4	2, 3	vtoclga 2664	. 2 |- ( A e. D -> C e. V )
5		fvmpt3.b	. . 3 |- F = ( x e. D |-> B )
6	1, 5	fvmptg 5269	. 2 |- ( ( A e. D /\ C e. V ) -> ( F ` A ) = C )
7	4, 6	mpdan 412	1 |- ( A e. D -> ( F ` A ) = C )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   |-> cmpt 3839   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930

This theorem is referenced by:  fvmpt3i  5273  frec2uzzd  9402  frec2uzsucd  9403