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Theorem afvvfunressn 43686
 Description: If the function value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvvfunressn ((𝐹'''𝐴) ∈ 𝐵 → Fun (𝐹 ↾ {𝐴}))

Proof of Theorem afvvfunressn
StepHypRef Expression
1 nfunsnafv 43685 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)
2 nvelim 43666 . . 3 ((𝐹'''𝐴) = V → ¬ (𝐹'''𝐴) ∈ 𝐵)
31, 2syl 17 . 2 (¬ Fun (𝐹 ↾ {𝐴}) → ¬ (𝐹'''𝐴) ∈ 𝐵)
43con4i 114 1 ((𝐹'''𝐴) ∈ 𝐵 → Fun (𝐹 ↾ {𝐴}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2112  Vcvv 3444  {csn 4528   ↾ cres 5525  Fun wfun 6322  '''cafv 43660 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-res 5535  df-iota 6287  df-fun 6330  df-fv 6336  df-aiota 43629  df-dfat 43662  df-afv 43663 This theorem is referenced by:  aovvfunressn  43730
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