Theorem fmptdf 6881

Description: A version of fmptd 6878 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 415	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶))
4	1, 3	ralrimi 3216	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
5		fmptdf.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fmpt 6874	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
7	4, 6	sylib 220	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  ∀wral 3138   ↦ cmpt 5146  ⟶wf 6351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363

This theorem is referenced by:  gsumesum  31318  voliune  31488  sdclem2  35032  fmptd2f  41525  limsupubuzmpt  42020  xlimmnfmpt  42144  xlimpnfmpt  42145  cncfiooicclem1  42196  dvnprodlem1  42251  stoweidlem35  42340  stoweidlem42  42347  stoweidlem48  42353  stirlinglem8  42386  sge0revalmpt  42680  sge0f1o  42684  sge0gerpmpt  42704  sge0ssrempt  42707  sge0ltfirpmpt  42710  sge0lempt  42712  sge0splitmpt  42713  sge0ss  42714  sge0rernmpt  42724  sge0lefimpt  42725  sge0clmpt  42727  sge0ltfirpmpt2  42728  sge0isummpt  42732  sge0xadd  42737  sge0fsummptf  42738  sge0snmptf  42739  sge0ge0mpt  42740  sge0repnfmpt  42741  sge0pnffigtmpt  42742  sge0gtfsumgt  42745  sge0pnfmpt  42747  meadjiun  42768  meaiunlelem  42770  omeiunle  42819  omeiunlempt  42822  opnvonmbllem1  42934  hoimbl2  42967  vonhoire  42974  vonn0ioo2  42992  vonn0icc2  42994  pimgtmnf  43020  issmfdmpt  43045  smfconst  43046  smfadd  43061  smfpimcclem  43101  smflimmpt  43104  smfsupmpt  43109  smfinfmpt  43113  smflimsuplem2  43115  gsumsplit2f  44107  fsuppmptdmf  44449