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Theorem fmptdf 7118
Description: A version of fmptd 7115 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 413 . . 3 (𝜑 → (𝑥𝐴𝐵𝐶))
41, 3ralrimi 3254 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
5 fmptdf.3 . . 3 𝐹 = (𝑥𝐴𝐵)
65fmpt 7111 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
74, 6sylib 217 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wnf 1785  wcel 2106  wral 3061  cmpt 5231  wf 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6545  df-fn 6546  df-f 6547
This theorem is referenced by:  elrspunidl  32808  gsumesum  33343  voliune  33513  sdclem2  36913  fmptd2f  44236  limsupubuzmpt  44734  xlimmnfmpt  44858  xlimpnfmpt  44859  cncfiooicclem1  44908  dvnprodlem1  44961  stoweidlem35  45050  stoweidlem42  45057  stoweidlem48  45063  stirlinglem8  45096  sge0revalmpt  45393  sge0f1o  45397  sge0gerpmpt  45417  sge0ssrempt  45420  sge0ltfirpmpt  45423  sge0lempt  45425  sge0splitmpt  45426  sge0ss  45427  sge0rernmpt  45437  sge0lefimpt  45438  sge0clmpt  45440  sge0ltfirpmpt2  45441  sge0isummpt  45445  sge0xadd  45450  sge0fsummptf  45451  sge0snmptf  45452  sge0ge0mpt  45453  sge0repnfmpt  45454  sge0pnffigtmpt  45455  sge0gtfsumgt  45458  sge0pnfmpt  45460  meadjiun  45481  meaiunlelem  45483  omeiunle  45532  omeiunlempt  45535  opnvonmbllem1  45647  hoimbl2  45680  vonhoire  45687  vonn0ioo2  45705  vonn0icc2  45707  issmfdmpt  45763  smfconst  45764  smfadd  45780  smfpimcclem  45822  smflimmpt  45825  smfinfmpt  45834  smflimsuplem2  45836  gsumsplit2f  46857  fsuppmptdmf  47146
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