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Theorem fmptdf 6342
Description: A version of fmptd 6340 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 450 . . 3 (𝜑 → (𝑥𝐴𝐵𝐶))
41, 3ralrimi 2951 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
5 fmptdf.3 . . 3 𝐹 = (𝑥𝐴𝐵)
65fmpt 6337 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
74, 6sylib 208 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wnf 1705  wcel 1987  wral 2907  cmpt 4673  wf 5843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855
This theorem is referenced by:  gsumesum  29902  voliune  30073  sdclem2  33170  fmptd2f  38917  limsupubuzmpt  39355  cncfiooicclem1  39410  dvnprodlem1  39467  stoweidlem35  39559  stoweidlem42  39566  stoweidlem48  39572  stirlinglem8  39605  sge0z  39899  sge0revalmpt  39902  sge0f1o  39906  sge0gerpmpt  39926  sge0ssrempt  39929  sge0ltfirpmpt  39932  sge0lempt  39934  sge0splitmpt  39935  sge0ss  39936  sge0rernmpt  39946  sge0lefimpt  39947  sge0clmpt  39949  sge0ltfirpmpt2  39950  sge0isummpt  39954  sge0xadd  39959  sge0fsummptf  39960  sge0snmptf  39961  sge0ge0mpt  39962  sge0repnfmpt  39963  sge0pnffigtmpt  39964  sge0gtfsumgt  39967  sge0pnfmpt  39969  meadjiun  39990  meaiunlelem  39992  omeiunle  40038  omeiunlempt  40041  opnvonmbllem1  40153  hoimbl2  40186  vonhoire  40193  vonn0ioo2  40211  vonn0icc2  40213  pimgtmnf  40239  issmfdmpt  40264  smfconst  40265  smfadd  40280  smfpimcclem  40320  smflimmpt  40323  smfsupmpt  40328  smfinfmpt  40332  smflimsuplem2  40334  gsumsplit2f  41431  fsuppmptdmf  41450
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