Theorem fvmpt2d 5605

Description: Deduction version of fvmpt2 5602. (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 5519	. . 3 |- ( ph -> ( F ` x ) = ( ( x e. A |-> B ) ` x ) )
3	2	adantr 276	. 2 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( ( x e. A |-> B ) ` x ) )
4		simpr 110	. . 3 |- ( ( ph /\ x e. A ) -> x e. A )
5		fvmpt2d.4	. . 3 |- ( ( ph /\ x e. A ) -> B e. V )
6		eqid 2177	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
7	6	fvmpt2 5602	. . 3 |- ( ( x e. A /\ B e. V ) -> ( ( x e. A |-> B ) ` x ) = B )
8	4, 5, 7	syl2anc 411	. 2 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> B ) ` x ) = B )
9	3, 8	eqtrd 2210	1 |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   |-> cmpt 4066   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226

This theorem is referenced by:  iseqf1olemjpcl  10498  iseqf1olemqpcl  10499  isumshft  11501  bj-charfun  14720  bj-charfundc  14721