Theorem fvmptd2 5758

Description: Deduction version of fvmpt 5753 (where the definition of the mapping does not depend on the common antecedent ph). (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
fvmptd2.1	|- F = ( x e. D |-> B )
fvmptd2.2	|- ( ( ph /\ x = A ) -> B = C )
fvmptd2.3	|- ( ph -> A e. D )
fvmptd2.4	|- ( ph -> C e. V )

Assertion

Ref	Expression
fvmptd2	|- ( ph -> ( F ` A ) = C )

Distinct variable groups:   x, A   x, C   x, D   ph, x

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

Proof of Theorem fvmptd2

Step	Hyp	Ref	Expression
1		fvmptd2.1	. . 3 |- F = ( x e. D |-> B )
2	1	a1i 9	. 2 |- ( ph -> F = ( x e. D |-> B ) )
3		fvmptd2.2	. 2 |- ( ( ph /\ x = A ) -> B = C )
4		fvmptd2.3	. 2 |- ( ph -> A e. D )
5		fvmptd2.4	. 2 |- ( ph -> C e. V )
6	2, 3, 4, 5	fvmptd 5757	1 |- ( ph -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   |-> cmpt 4170   ` cfv 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-iota 5311  df-fun 5353  df-fv 5359

This theorem is referenced by:  gausslemma2dlem2  15927  gausslemma2dlem3  15928  vtxdgfval  16275