Theorem fvmptdv 5647

Description: Alternate deduction version of fvmpt 5635, 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 2336	. 2 |- F/_ x F
5		nfv 1539	. 2 |- F/ x ps
6	1, 2, 3, 4, 5	fvmptdf 5646	1 |- ( ph -> ( F = ( x e. D |-> B ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   |-> cmpt 4091   ` cfv 5255

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 4148  ax-pow 4204  ax-pr 4239

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 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263

This theorem is referenced by: (None)