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Theorem nfunsnaov 39915
 Description: If the restriction of a class to a singleton is not a function, its operation value is the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
nfunsnaov (¬ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩}) → ((𝐴𝐹𝐵)) = V)

Proof of Theorem nfunsnaov
StepHypRef Expression
1 df-aov 39847 . 2 ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2 nfunsnafv 39871 . 2 (¬ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩}) → (𝐹'''⟨𝐴, 𝐵⟩) = V)
31, 2syl5eq 2656 1 (¬ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩}) → ((𝐴𝐹𝐵)) = V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475  Vcvv 3173  {csn 4125  ⟨cop 4131   ↾ cres 5040  Fun wfun 5798  '''cafv 39843   ((caov 39844 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-un 3545  df-if 4037  df-fv 5812  df-dfat 39845  df-afv 39846  df-aov 39847 This theorem is referenced by: (None)
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