Theorem fmptdf 6387

Description: A version of fmptd 6385 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 450	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶))
4	1, 3	ralrimi 2957	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
5		fmptdf.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fmpt 6381	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
7	4, 6	sylib 208	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  ∀wral 2912   ↦ cmpt 4729  ⟶wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  gsumesum  30121  voliune  30292  sdclem2  33538  fmptd2f  39442  limsupubuzmpt  39951  xlimmnfmpt  40069  xlimpnfmpt  40070  cncfiooicclem1  40106  dvnprodlem1  40161  stoweidlem35  40252  stoweidlem42  40259  stoweidlem48  40265  stirlinglem8  40298  sge0z  40592  sge0revalmpt  40595  sge0f1o  40599  sge0gerpmpt  40619  sge0ssrempt  40622  sge0ltfirpmpt  40625  sge0lempt  40627  sge0splitmpt  40628  sge0ss  40629  sge0rernmpt  40639  sge0lefimpt  40640  sge0clmpt  40642  sge0ltfirpmpt2  40643  sge0isummpt  40647  sge0xadd  40652  sge0fsummptf  40653  sge0snmptf  40654  sge0ge0mpt  40655  sge0repnfmpt  40656  sge0pnffigtmpt  40657  sge0gtfsumgt  40660  sge0pnfmpt  40662  meadjiun  40683  meaiunlelem  40685  omeiunle  40731  omeiunlempt  40734  opnvonmbllem1  40846  hoimbl2  40879  vonhoire  40886  vonn0ioo2  40904  vonn0icc2  40906  pimgtmnf  40932  issmfdmpt  40957  smfconst  40958  smfadd  40973  smfpimcclem  41013  smflimmpt  41016  smfsupmpt  41021  smfinfmpt  41025  smflimsuplem2  41027  gsumsplit2f  42143  fsuppmptdmf  42162