Theorem fvmptdv 5766

Description: Alternate deduction version of fvmpt 5754, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

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 ) )

Assertion

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

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

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

Proof of Theorem fvmptdv

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		nfcv 2384	. 2 |- F/_ x F
5		nfv 1577	. 2 |- F/ x ps
6	1, 2, 3, 4, 5	fvmptdf 5765	1 |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   |-> cmpt 4171   ` cfv 5352

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 4228  ax-pow 4287  ax-pr 4322

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 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360

This theorem is referenced by: (None)