Theorem fvmpt2d 5720

Description: Deduction version of fvmpt2 5717. (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 5628	. . 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 2229	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
7	6	fvmpt2 5717	. . 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 2262	1 |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   |-> cmpt 4144   ` cfv 5317

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 4201  ax-pow 4257  ax-pr 4292

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 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325

This theorem is referenced by:  iseqf1olemjpcl  10725  iseqf1olemqpcl  10726  isumshft  11996  bj-charfun  16128  bj-charfundc  16129