Theorem fvmptdf 6296

Description: Alternate deduction version of fvmpt 6282, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) (Proof shortened by AV, 19-Jan-2022.)

Hypotheses

Ref	Expression
fvmptdf.1	|- ( ph -> A e. D )
fvmptdf.2	|- ( ( ph /\ x = A ) -> B e. V )
fvmptdf.3	|- ( ( ph /\ x = A ) -> ( ( F ` A ) = B -> ps ) )
fvmptdf.4	|- F/_ x F
fvmptdf.5	|- F/ x ps

Assertion

Ref	Expression
fvmptdf	|- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )

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

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

Proof of Theorem fvmptdf

Step	Hyp	Ref	Expression
1		fvmptdf.1	. 2 |- ( ph -> A e. D )
2		fvmptdf.2	. 2 |- ( ( ph /\ x = A ) -> B e. V )
3		fvmptdf.3	. 2 |- ( ( ph /\ x = A ) -> ( ( F ` A ) = B -> ps ) )
4		fvmptdf.4	. 2 |- F/_ x F
5		fvmptdf.5	. 2 |- F/ x ps
6		nfv 1843	. 2 |- F/ x ph
7	1, 2, 3, 4, 5, 6	fvmptd3f 6295	1 |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   |-> cmpt 4729   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  fvmptdv  6297  yonedalem4b  16916