Theorem fmptdf 7136

Description: A version of fmptd 7133 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 412	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶))
4	1, 3	ralrimi 3254	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
5		fmptdf.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fmpt 7129	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
7	4, 6	sylib 218	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536  Ⅎwnf 1779   ∈ wcel 2105  ∀wral 3058   ↦ cmpt 5230  ⟶wf 6558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-fun 6564  df-fn 6565  df-f 6566

This theorem is referenced by:  elrspunidl  33435  gsumesum  34039  voliune  34209  sdclem2  37728  fmptd2f  45177  limsupubuzmpt  45674  xlimmnfmpt  45798  xlimpnfmpt  45799  cncfiooicclem1  45848  stoweidlem35  45990  stoweidlem42  45997  stoweidlem48  46003  stirlinglem8  46036  sge0revalmpt  46333  sge0gerpmpt  46357  sge0ssrempt  46360  sge0ltfirpmpt  46363  sge0lempt  46365  sge0splitmpt  46366  sge0ss  46367  sge0rernmpt  46377  sge0lefimpt  46378  sge0clmpt  46380  sge0ltfirpmpt2  46381  sge0isummpt  46385  sge0xadd  46390  sge0fsummptf  46391  sge0snmptf  46392  sge0ge0mpt  46393  sge0repnfmpt  46394  sge0pnffigtmpt  46395  sge0gtfsumgt  46398  sge0pnfmpt  46400  meadjiun  46421  meaiunlelem  46423  omeiunle  46472  omeiunlempt  46475  opnvonmbllem1  46587  hoimbl2  46620  vonhoire  46627  vonn0ioo2  46645  vonn0icc2  46647  issmfdmpt  46703  smfconst  46704  smfadd  46720  smfpimcclem  46762  smflimmpt  46765  smflimsuplem2  46776  gsumsplit2f  48023  fsuppmptdmf  48222