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Theorem fmptdf 7117
Description: A version of fmptd 7114 using bound-variable hypothesis instead of a distinct variable condition for 𝜑. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypotheses
Ref Expression
fmptdf.1 𝑥𝜑
fmptdf.2 ((𝜑𝑥𝐴) → 𝐵𝐶)
fmptdf.3 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fmptdf (𝜑𝐹:𝐴𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fmptdf
StepHypRef Expression
1 fmptdf.1 . . 3 𝑥𝜑
2 fmptdf.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵𝐶)
32ex 414 . . 3 (𝜑 → (𝑥𝐴𝐵𝐶))
41, 3ralrimi 3255 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
5 fmptdf.3 . . 3 𝐹 = (𝑥𝐴𝐵)
65fmpt 7110 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
74, 6sylib 217 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wnf 1786  wcel 2107  wral 3062  cmpt 5232  wf 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547  df-f 6548
This theorem is referenced by:  elrspunidl  32546  gsumesum  33057  voliune  33227  sdclem2  36610  fmptd2f  43937  limsupubuzmpt  44435  xlimmnfmpt  44559  xlimpnfmpt  44560  cncfiooicclem1  44609  dvnprodlem1  44662  stoweidlem35  44751  stoweidlem42  44758  stoweidlem48  44764  stirlinglem8  44797  sge0revalmpt  45094  sge0f1o  45098  sge0gerpmpt  45118  sge0ssrempt  45121  sge0ltfirpmpt  45124  sge0lempt  45126  sge0splitmpt  45127  sge0ss  45128  sge0rernmpt  45138  sge0lefimpt  45139  sge0clmpt  45141  sge0ltfirpmpt2  45142  sge0isummpt  45146  sge0xadd  45151  sge0fsummptf  45152  sge0snmptf  45153  sge0ge0mpt  45154  sge0repnfmpt  45155  sge0pnffigtmpt  45156  sge0gtfsumgt  45159  sge0pnfmpt  45161  meadjiun  45182  meaiunlelem  45184  omeiunle  45233  omeiunlempt  45236  opnvonmbllem1  45348  hoimbl2  45381  vonhoire  45388  vonn0ioo2  45406  vonn0icc2  45408  issmfdmpt  45464  smfconst  45465  smfadd  45481  smfpimcclem  45523  smflimmpt  45526  smfinfmpt  45535  smflimsuplem2  45537  gsumsplit2f  46590  fsuppmptdmf  47057
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