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Theorem fmptdf 6858
 Description: A version of fmptd 6855 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 416 . . 3 (𝜑 → (𝑥𝐴𝐵𝐶))
41, 3ralrimi 3180 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
5 fmptdf.3 . . 3 𝐹 = (𝑥𝐴𝐵)
65fmpt 6851 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
74, 6sylib 221 1 (𝜑𝐹:𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  ∀wral 3106   ↦ cmpt 5110  ⟶wf 6320 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332 This theorem is referenced by:  elrspunidl  31021  gsumesum  31440  voliune  31610  sdclem2  35196  fmptd2f  41886  limsupubuzmpt  42376  xlimmnfmpt  42500  xlimpnfmpt  42501  cncfiooicclem1  42550  dvnprodlem1  42603  stoweidlem35  42692  stoweidlem42  42699  stoweidlem48  42705  stirlinglem8  42738  sge0revalmpt  43032  sge0f1o  43036  sge0gerpmpt  43056  sge0ssrempt  43059  sge0ltfirpmpt  43062  sge0lempt  43064  sge0splitmpt  43065  sge0ss  43066  sge0rernmpt  43076  sge0lefimpt  43077  sge0clmpt  43079  sge0ltfirpmpt2  43080  sge0isummpt  43084  sge0xadd  43089  sge0fsummptf  43090  sge0snmptf  43091  sge0ge0mpt  43092  sge0repnfmpt  43093  sge0pnffigtmpt  43094  sge0gtfsumgt  43097  sge0pnfmpt  43099  meadjiun  43120  meaiunlelem  43122  omeiunle  43171  omeiunlempt  43174  opnvonmbllem1  43286  hoimbl2  43319  vonhoire  43326  vonn0ioo2  43344  vonn0icc2  43346  pimgtmnf  43372  issmfdmpt  43397  smfconst  43398  smfadd  43413  smfpimcclem  43453  smflimmpt  43456  smfsupmpt  43461  smfinfmpt  43465  smflimsuplem2  43467  gsumsplit2f  44455  fsuppmptdmf  44798
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