Theorem fmptdf 7151

Description: A version of fmptd 7148 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 3263	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
5		fmptdf.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fmpt 7144	. 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 1537  Ⅎwnf 1781   ∈ wcel 2108  ∀wral 3067   ↦ cmpt 5249  ⟶wf 6569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-fun 6575  df-fn 6576  df-f 6577

This theorem is referenced by:  elrspunidl  33421  gsumesum  34023  voliune  34193  sdclem2  37702  fmptd2f  45142  limsupubuzmpt  45640  xlimmnfmpt  45764  xlimpnfmpt  45765  cncfiooicclem1  45814  dvnprodlem1  45867  stoweidlem35  45956  stoweidlem42  45963  stoweidlem48  45969  stirlinglem8  46002  sge0revalmpt  46299  sge0f1o  46303  sge0gerpmpt  46323  sge0ssrempt  46326  sge0ltfirpmpt  46329  sge0lempt  46331  sge0splitmpt  46332  sge0ss  46333  sge0rernmpt  46343  sge0lefimpt  46344  sge0clmpt  46346  sge0ltfirpmpt2  46347  sge0isummpt  46351  sge0xadd  46356  sge0fsummptf  46357  sge0snmptf  46358  sge0ge0mpt  46359  sge0repnfmpt  46360  sge0pnffigtmpt  46361  sge0gtfsumgt  46364  sge0pnfmpt  46366  meadjiun  46387  meaiunlelem  46389  omeiunle  46438  omeiunlempt  46441  opnvonmbllem1  46553  hoimbl2  46586  vonhoire  46593  vonn0ioo2  46611  vonn0icc2  46613  issmfdmpt  46669  smfconst  46670  smfadd  46686  smfpimcclem  46728  smflimmpt  46731  smfinfmpt  46740  smflimsuplem2  46742  gsumsplit2f  47903  fsuppmptdmf  48106