Theorem fmptdf 7063

Description: A version of fmptd 7060 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 3236	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
5		fmptdf.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fmpt 7056	. 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 1542  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3052   ↦ cmpt 5167  ⟶wf 6488

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-fun 6494  df-fn 6495  df-f 6496

This theorem is referenced by:  elrspunidl  33503  gsumesum  34219  voliune  34389  sdclem2  38077  fmptd2f  45682  limsupubuzmpt  46165  xlimmnfmpt  46289  xlimpnfmpt  46290  cncfiooicclem1  46339  stoweidlem35  46481  stoweidlem42  46488  stoweidlem48  46494  stirlinglem8  46527  sge0revalmpt  46824  sge0gerpmpt  46848  sge0ssrempt  46851  sge0ltfirpmpt  46854  sge0lempt  46856  sge0splitmpt  46857  sge0ss  46858  sge0rernmpt  46868  sge0lefimpt  46869  sge0clmpt  46871  sge0ltfirpmpt2  46872  sge0isummpt  46876  sge0xadd  46881  sge0fsummptf  46882  sge0snmptf  46883  sge0ge0mpt  46884  sge0repnfmpt  46885  sge0pnffigtmpt  46886  sge0gtfsumgt  46889  sge0pnfmpt  46891  meadjiun  46912  meaiunlelem  46914  omeiunle  46963  omeiunlempt  46966  opnvonmbllem1  47078  hoimbl2  47111  vonhoire  47118  vonn0ioo2  47136  vonn0icc2  47138  issmfdmpt  47194  smfconst  47195  smfadd  47211  smfpimcclem  47253  smflimmpt  47256  smflimsuplem2  47267  gsumsplit2f  48668  fsuppmptdmf  48866