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Theorem afvnfundmuv 41743
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 41736 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
2 iffalse 4239 . 2 𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V)
31, 2syl5eq 2806 1 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1632  Vcvv 3340  ifcif 4230  cfv 6049   defAt wdfat 41717  '''cafv 41718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-un 3720  df-if 4231  df-fv 6057  df-afv 41721
This theorem is referenced by:  ndmafv  41744  nfunsnafv  41746  afvnufveq  41751  afvres  41776  afvco2  41780  aovnfundmuv  41786
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