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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 412 . . 3 (𝜑 → (𝑥𝐴𝐵𝐶))
41, 3ralrimi 3253 . 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 395   = wceq 1540  wnf 1784  wcel 2105  wral 3060  cmpt 5231  wf 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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  32986  gsumesum  33521  voliune  33691  sdclem2  37074  fmptd2f  44396  limsupubuzmpt  44894  xlimmnfmpt  45018  xlimpnfmpt  45019  cncfiooicclem1  45068  dvnprodlem1  45121  stoweidlem35  45210  stoweidlem42  45217  stoweidlem48  45223  stirlinglem8  45256  sge0revalmpt  45553  sge0f1o  45557  sge0gerpmpt  45577  sge0ssrempt  45580  sge0ltfirpmpt  45583  sge0lempt  45585  sge0splitmpt  45586  sge0ss  45587  sge0rernmpt  45597  sge0lefimpt  45598  sge0clmpt  45600  sge0ltfirpmpt2  45601  sge0isummpt  45605  sge0xadd  45610  sge0fsummptf  45611  sge0snmptf  45612  sge0ge0mpt  45613  sge0repnfmpt  45614  sge0pnffigtmpt  45615  sge0gtfsumgt  45618  sge0pnfmpt  45620  meadjiun  45641  meaiunlelem  45643  omeiunle  45692  omeiunlempt  45695  opnvonmbllem1  45807  hoimbl2  45840  vonhoire  45847  vonn0ioo2  45865  vonn0icc2  45867  issmfdmpt  45923  smfconst  45924  smfadd  45940  smfpimcclem  45982  smflimmpt  45985  smfinfmpt  45994  smflimsuplem2  45996  gsumsplit2f  47017  fsuppmptdmf  47220
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