Theorem fmptdf 5812

Description: A version of fmptd 5809 using bound-variable hypothesis instead of a distinct variable condition for ph. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Hypotheses

Ref	Expression
fmptdf.1	|- F/ x ph
fmptdf.2	|- ( ( ph /\ x e. A ) -> B e. C )
fmptdf.3	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
fmptdf	|- ( ph -> F : A --> C )

Distinct variable groups:   x, A   x, C

Allowed substitution hints:   ph( x)   B( x)   F( x)

Proof of Theorem fmptdf

Step	Hyp	Ref	Expression
1		fmptdf.1	. . 3 |- F/ x ph
2		fmptdf.2	. . . 4 |- ( ( ph /\ x e. A ) -> B e. C )
3	2	ex 115	. . 3 |- ( ph -> ( x e. A -> B e. C ) )
4	1, 3	ralrimi 2604	. 2 |- ( ph -> A. x e. A B e. C )
5		fmptdf.3	. . 3 |- F = ( x e. A |-> B )
6	5	fmpt 5805	. 2 |- ( A. x e. A B e. C <-> F : A --> C )
7	4, 6	sylib 122	1 |- ( ph -> F : A --> C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   F/wnf 1509   e. wcel 2202   A.wral 2511   |-> cmpt 4155   -->wf 5329

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341

This theorem is referenced by:  mkvprop  7417