Theorem fmptdf 7116

Description: A version of fmptd 7113 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 418	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶))
4	1, 3	ralrimi 3265	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
5		fmptdf.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fmpt 7109	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
7	4, 6	sylib 221	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2146  ∀wral 3081   ↦ cmpt 5194  ⟶wf 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-fun 6542  df-fn 6543  df-f 6544

This theorem is used by:  elrspunidl  33776  gsumesum  34489  voliune  34660  sdclem2  38426  fmptd2f  45983  limsupubuzmpt  46466  xlimmnfmpt  46590  xlimpnfmpt  46591  cncfiooicclem1  46640  stoweidlem35  46782  stoweidlem42  46789  stoweidlem48  46795  stirlinglem8  46828  sge0revalmpt  47125  sge0gerpmpt  47149  sge0ssrempt  47152  sge0ltfirpmpt  47155  sge0lempt  47157  sge0splitmpt  47158  sge0ss  47159  sge0rernmpt  47169  sge0lefimpt  47170  sge0clmpt  47172  sge0ltfirpmpt2  47173  sge0isummpt  47177  sge0xadd  47182  sge0fsummptf  47183  sge0snmptf  47184  sge0ge0mpt  47185  sge0repnfmpt  47186  sge0pnffigtmpt  47187  sge0gtfsumgt  47190  sge0pnfmpt  47192  meadjiun  47213  meaiunlelem  47215  omeiunle  47264  omeiunlempt  47267  opnvonmbllem1  47379  hoimbl2  47412  vonhoire  47419  vonn0ioo2  47437  vonn0icc2  47439  issmfdmpt  47495  smfconst  47496  smfadd  47512  smfpimcclem  47554  smflimmpt  47557  smflimsuplem2  47568  gsumsplit2f  48978  fsuppmptdmf  49191