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Theorem fmptdf 27589
Description: A version of fmptd 5856 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 424 . . 3  |-  ( ph  ->  ( x  e.  A  ->  B  e.  C ) )
41, 3ralrimi 2751 . 2  |-  ( ph  ->  A. x  e.  A  B  e.  C )
5 fmptdf.3 . . 3  |-  F  =  ( x  e.  A  |->  B )
65fmpt 5853 . 2  |-  ( A. x  e.  A  B  e.  C  <->  F : A --> C )
74, 6sylib 189 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   F/wnf 1550    = wceq 1649    e. wcel 1721   A.wral 2670    e. cmpt 4230   -->wf 5413
This theorem is referenced by:  stoweidlem35  27655  stoweidlem42  27662  stoweidlem48  27668  stirlinglem8  27701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425
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