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Theorem afvnfundmuv 40982
Description: If a set is not in the domain of a class or the class is not a function restricted to the set, then the function value for this set is the universe. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
afvnfundmuv 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)

Proof of Theorem afvnfundmuv
StepHypRef Expression
1 dfafv2 40975 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
2 iffalse 4086 . 2 𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V)
31, 2syl5eq 2666 1 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1481  Vcvv 3195  ifcif 4077  cfv 5876   defAt wdfat 40956  '''cafv 40957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-un 3572  df-if 4078  df-fv 5884  df-afv 40960
This theorem is referenced by:  ndmafv  40983  nfunsnafv  40985  afvnufveq  40990  afvres  41015  afvco2  41019  aovnfundmuv  41025
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