Theorem fvmptd2 5639

Description: Deduction version of fvmpt 5634 (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 5638	1 |- ( ph -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   |-> cmpt 4090   ` cfv 5254

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262

This theorem is referenced by:  gausslemma2dlem2  15178  gausslemma2dlem3  15179