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Theorem fmptdf 6619
Description: A version of fmptd 6616 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 399 . . 3 (𝜑 → (𝑥𝐴𝐵𝐶))
41, 3ralrimi 3156 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
5 fmptdf.3 . . 3 𝐹 = (𝑥𝐴𝐵)
65fmpt 6612 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
74, 6sylib 209 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wnf 1863  wcel 2157  wral 3107  cmpt 4934  wf 6107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pr 5109
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-fv 6119
This theorem is referenced by:  gsumesum  30469  voliune  30640  sdclem2  33868  fmptd2f  39944  limsupubuzmpt  40449  xlimmnfmpt  40567  xlimpnfmpt  40568  cncfiooicclem1  40604  dvnprodlem1  40659  stoweidlem35  40749  stoweidlem42  40756  stoweidlem48  40762  stirlinglem8  40795  sge0z  41089  sge0revalmpt  41092  sge0f1o  41096  sge0gerpmpt  41116  sge0ssrempt  41119  sge0ltfirpmpt  41122  sge0lempt  41124  sge0splitmpt  41125  sge0ss  41126  sge0rernmpt  41136  sge0lefimpt  41137  sge0clmpt  41139  sge0ltfirpmpt2  41140  sge0isummpt  41144  sge0xadd  41149  sge0fsummptf  41150  sge0snmptf  41151  sge0ge0mpt  41152  sge0repnfmpt  41153  sge0pnffigtmpt  41154  sge0gtfsumgt  41157  sge0pnfmpt  41159  meadjiun  41180  meaiunlelem  41182  omeiunle  41231  omeiunlempt  41234  opnvonmbllem1  41346  hoimbl2  41379  vonhoire  41386  vonn0ioo2  41404  vonn0icc2  41406  pimgtmnf  41432  issmfdmpt  41457  smfconst  41458  smfadd  41473  smfpimcclem  41513  smflimmpt  41516  smfsupmpt  41521  smfinfmpt  41525  smflimsuplem2  41527  gsumsplit2f  42729  fsuppmptdmf  42748
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