Theorem fvmpt2d 5404

Description: Deduction version of fvmpt2 5401. (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 5322	. . 3 |- ( ph -> ( F ` x ) = ( ( x e. A |-> B ) ` x ) )
3	2	adantr 271	. 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 2089	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
7	6	fvmpt2 5401	. . 3 |- ( ( x e. A /\ B e. V ) -> ( ( x e. A |-> B ) ` x ) = B )
8	4, 5, 7	syl2anc 404	. 2 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> B ) ` x ) = B )
9	3, 8	eqtrd 2121	1 |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439   |-> cmpt 3907   ` cfv 5030

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-sbc 2844  df-csb 2937  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-mpt 3909  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-iota 4995  df-fun 5032  df-fv 5038

This theorem is referenced by:  iseqf1olemjpcl  9987  iseqf1olemqpcl  9988  isumshft  10947