Theorem fvmpt2d 5515

Description: Deduction version of fvmpt2 5512. (Contributed by Thierry Arnoux, 8-Dec-2016.)

Hypotheses

Ref	Expression
fvmpt2d.1	|- ( ph -> F = ( x e. A |-> B ) )
fvmpt2d.4	|- ( ( ph /\ x e. A ) -> B e. V )

Assertion

Ref	Expression
fvmpt2d	|- ( ( ph /\ x e. A ) -> ( F ` x ) = B )

Distinct variable group:   x, A

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

Proof of Theorem fvmpt2d

Step	Hyp	Ref	Expression
1		fvmpt2d.1	. . . 4 |- ( ph -> F = ( x e. A |-> B ) )
2	1	fveq1d 5431	. . 3 |- ( ph -> ( F ` x ) = ( ( x e. A |-> B ) ` x ) )
3	2	adantr 274	. 2 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( ( x e. A |-> B ) ` x ) )
4		simpr 109	. . 3 |- ( ( ph /\ x e. A ) -> x e. A )
5		fvmpt2d.4	. . 3 |- ( ( ph /\ x e. A ) -> B e. V )
6		eqid 2140	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
7	6	fvmpt2 5512	. . 3 |- ( ( x e. A /\ B e. V ) -> ( ( x e. A |-> B ) ` x ) = B )
8	4, 5, 7	syl2anc 409	. 2 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> B ) ` x ) = B )
9	3, 8	eqtrd 2173	1 |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   |-> cmpt 3997   ` cfv 5131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139

This theorem is referenced by:  iseqf1olemjpcl  10299  iseqf1olemqpcl  10300  isumshft  11291