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Theorem fmptdf 27708
Description: A version of fmptd 5896 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
StepHypRef Expression
1 fmptdf.1 . . 3  |-  F/ x ph
2 fmptdf.2 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  C )
32ex 425 . . 3  |-  ( ph  ->  ( x  e.  A  ->  B  e.  C ) )
41, 3ralrimi 2789 . 2  |-  ( ph  ->  A. x  e.  A  B  e.  C )
5 fmptdf.3 . . 3  |-  F  =  ( x  e.  A  |->  B )
65fmpt 5893 . 2  |-  ( A. x  e.  A  B  e.  C  <->  F : A --> C )
74, 6sylib 190 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   F/wnf 1554    = wceq 1653    e. wcel 1726   A.wral 2707    e. cmpt 4269   -->wf 5453
This theorem is referenced by:  stoweidlem35  27774  stoweidlem42  27781  stoweidlem48  27787  stirlinglem8  27820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465
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