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Theorem fmptdf 27041
Description: A version of fmptd 5764 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 423 . . 3  |-  ( ph  ->  ( x  e.  A  ->  B  e.  C ) )
41, 3ralrimi 2700 . 2  |-  ( ph  ->  A. x  e.  A  B  e.  C )
5 fmptdf.3 . . 3  |-  F  =  ( x  e.  A  |->  B )
65fmpt 5761 . 2  |-  ( A. x  e.  A  B  e.  C  <->  F : A --> C )
74, 6sylib 188 1  |-  ( ph  ->  F : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   F/wnf 1544    = wceq 1642    e. wcel 1710   A.wral 2619    e. cmpt 4156   -->wf 5330
This theorem is referenced by:  stirlinglem8  27153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fv 5342
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