Theorem fvmpt2d 5679

Description: Deduction version of fvmpt2 5676. (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 5591	. . 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 2206	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
7	6	fvmpt2 5676	. . 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 2239	1 |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   |-> cmpt 4113   ` cfv 5280

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-iota 5241  df-fun 5282  df-fv 5288

This theorem is referenced by:  iseqf1olemjpcl  10675  iseqf1olemqpcl  10676  isumshft  11876  bj-charfun  15881  bj-charfundc  15882