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Theorem fmptdf 6037
Description: A version of fmptd 6036 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 434 . . 3  |-  ( ph  ->  ( x  e.  A  ->  B  e.  C ) )
41, 3ralrimi 2857 . 2  |-  ( ph  ->  A. x  e.  A  B  e.  C )
5 fmptdf.3 . . 3  |-  F  =  ( x  e.  A  |->  B )
65fmpt 6033 . 2  |-  ( A. x  e.  A  B  e.  C  <->  F : A --> C )
74, 6sylib 196 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374   F/wnf 1594    e. wcel 1762   A.wral 2807    |-> cmpt 4498   -->wf 5575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587
This theorem is referenced by:  gsumesum  27693  stoweidlem35  31290  stoweidlem42  31297  stoweidlem48  31303  stirlinglem8  31336  gsumsplit2f  31894  fsuppmptdmf  31922
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