Theorem fmptdf 6550

Description: A version of fmptd 6548 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 449	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶))
4	1, 3	ralrimi 3095	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
5		fmptdf.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fmpt 6544	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
7	4, 6	sylib 208	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632  Ⅎwnf 1857   ∈ wcel 2139  ∀wral 3050   ↦ cmpt 4881  ⟶wf 6045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057

This theorem is referenced by:  gsumesum  30430  voliune  30601  sdclem2  33851  fmptd2f  39941  limsupubuzmpt  40454  xlimmnfmpt  40572  xlimpnfmpt  40573  cncfiooicclem1  40609  dvnprodlem1  40664  stoweidlem35  40755  stoweidlem42  40762  stoweidlem48  40768  stirlinglem8  40801  sge0z  41095  sge0revalmpt  41098  sge0f1o  41102  sge0gerpmpt  41122  sge0ssrempt  41125  sge0ltfirpmpt  41128  sge0lempt  41130  sge0splitmpt  41131  sge0ss  41132  sge0rernmpt  41142  sge0lefimpt  41143  sge0clmpt  41145  sge0ltfirpmpt2  41146  sge0isummpt  41150  sge0xadd  41155  sge0fsummptf  41156  sge0snmptf  41157  sge0ge0mpt  41158  sge0repnfmpt  41159  sge0pnffigtmpt  41160  sge0gtfsumgt  41163  sge0pnfmpt  41165  meadjiun  41186  meaiunlelem  41188  omeiunle  41237  omeiunlempt  41240  opnvonmbllem1  41352  hoimbl2  41385  vonhoire  41392  vonn0ioo2  41410  vonn0icc2  41412  pimgtmnf  41438  issmfdmpt  41463  smfconst  41464  smfadd  41479  smfpimcclem  41519  smflimmpt  41522  smfsupmpt  41527  smfinfmpt  41531  smflimsuplem2  41533  gsumsplit2f  42653  fsuppmptdmf  42672