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Theorem nfunsnafv 41543
 Description: If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6263. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
nfunsnafv (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)

Proof of Theorem nfunsnafv
StepHypRef Expression
1 df-dfat 41517 . . . 4 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
21simprbi 479 . . 3 (𝐹 defAt 𝐴 → Fun (𝐹 ↾ {𝐴}))
32con3i 150 . 2 (¬ Fun (𝐹 ↾ {𝐴}) → ¬ 𝐹 defAt 𝐴)
4 afvnfundmuv 41540 . 2 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
53, 4syl 17 1 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1523   ∈ wcel 2030  Vcvv 3231  {csn 4210  dom cdm 5143   ↾ cres 5145  Fun wfun 5920   defAt wdfat 41514  '''cafv 41515 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-un 3612  df-if 4120  df-fv 5934  df-dfat 41517  df-afv 41518 This theorem is referenced by:  afvvfunressn  41544  nfunsnaov  41587
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