Theorem fvmptd2 5671

Description: Deduction version of fvmpt 5666 (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 5670	1 |- ( ph -> ( F ` A ) = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   |-> cmpt 4110   ` cfv 5277

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 4167  ax-pow 4223  ax-pr 4258

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 3001  df-csb 3096  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-iota 5238  df-fun 5279  df-fv 5285

This theorem is referenced by:  gausslemma2dlem2  15589  gausslemma2dlem3  15590