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Theorem fmptdf 6294
 Description: A version of fmptd 6292 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 449 . . 3 (𝜑 → (𝑥𝐴𝐵𝐶))
41, 3ralrimi 2940 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
5 fmptdf.3 . . 3 𝐹 = (𝑥𝐴𝐵)
65fmpt 6289 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
74, 6sylib 207 1 (𝜑𝐹:𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  ∀wral 2896   ↦ cmpt 4643  ⟶wf 5800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812 This theorem is referenced by:  gsumesum  29448  voliune  29619  sdclem2  32708  cncfiooicclem1  38779  dvnprodlem1  38836  stoweidlem35  38928  stoweidlem42  38935  stoweidlem48  38941  stirlinglem8  38974  sge0z  39268  sge0revalmpt  39271  sge0f1o  39275  sge0gerpmpt  39295  sge0ssrempt  39298  sge0ltfirpmpt  39301  sge0lempt  39303  sge0splitmpt  39304  sge0ss  39305  sge0rernmpt  39315  sge0lefimpt  39316  sge0clmpt  39318  sge0ltfirpmpt2  39319  sge0isummpt  39323  sge0xadd  39328  sge0fsummptf  39329  sge0snmptf  39330  sge0ge0mpt  39331  sge0repnfmpt  39332  sge0pnffigtmpt  39333  sge0gtfsumgt  39336  sge0pnfmpt  39338  meadjiun  39359  meaiunlelem  39361  omeiunle  39407  omeiunlempt  39410  opnvonmbllem1  39522  hoimbl2  39555  vonhoire  39563  vonn0ioo2  39581  vonn0icc2  39583  pimgtmnf  39609  issmfdmpt  39635  smfconst  39636  smfadd  39651  gsumsplit2f  41936  fsuppmptdmf  41956
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