Theorem fmptdf 7149

Description: A version of fmptd 7146 using bound-variable hypothesis instead of a distinct variable condition for 𝜑. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Hypotheses

Ref	Expression
fmptdf.1	⊢ Ⅎ𝑥𝜑
fmptdf.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
fmptdf.3	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
fmptdf	⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fmptdf

Step	Hyp	Ref	Expression
1		fmptdf.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		fmptdf.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
3	2	ex 418	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶))
4	1, 3	ralrimi 3264	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
5		fmptdf.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fmpt 7142	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
7	4, 6	sylib 221	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2147  ∀wral 3080   ↦ cmpt 5192  ⟶wf 6562

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-fun 6568  df-fn 6569  df-f 6570

This theorem is used by:  elrspunidl  34082  gsumesum  34802  voliune  34973  sdclem2  38942  fmptd2f  46520  limsupubuzmpt  47002  xlimmnfmpt  47126  xlimpnfmpt  47127  cncfiooicclem1  47176  stoweidlem35  47318  stoweidlem42  47325  stoweidlem48  47331  stirlinglem8  47364  sge0revalmpt  47661  sge0gerpmpt  47685  sge0ssrempt  47688  sge0ltfirpmpt  47691  sge0lempt  47693  sge0splitmpt  47694  sge0ss  47695  sge0rernmpt  47705  sge0lefimpt  47706  sge0clmpt  47708  sge0ltfirpmpt2  47709  sge0isummpt  47713  sge0xadd  47718  sge0fsummptf  47719  sge0snmptf  47720  sge0ge0mpt  47721  sge0repnfmpt  47722  sge0pnffigtmpt  47723  sge0gtfsumgt  47726  sge0pnfmpt  47728  meadjiun  47749  meaiunlelem  47751  omeiunle  47800  omeiunlempt  47803  opnvonmbllem1  47915  hoimbl2  47948  vonhoire  47955  vonn0ioo2  47973  vonn0icc2  47975  issmfdmpt  48031  smfconst  48032  smfadd  48048  smfpimcclem  48090  smflimmpt  48093  smflimsuplem2  48104  gsumsplit2f  49550  fsuppmptdmf  49821