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Theorem afvfundmfveq 41738
Description: If a class is a function restricted to a member of its domain, then the function value for this member is equal for both definitions. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvfundmfveq (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))

Proof of Theorem afvfundmfveq
StepHypRef Expression
1 dfafv2 41732 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
2 iftrue 4231 . 2 (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹𝐴), V) = (𝐹𝐴))
31, 2syl5eq 2817 1 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  Vcvv 3351  ifcif 4225  cfv 6031   defAt wdfat 41713  '''cafv 41714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-un 3728  df-if 4226  df-fv 6039  df-afv 41717
This theorem is referenced by:  afvnufveq  41747  afvfvn0fveq  41750  afv0nbfvbi  41751  afveu  41753  fnbrafvb  41754  afvelrn  41768  afvres  41772  tz6.12-afv  41773  dmfcoafv  41775  afvco2  41776  rlimdmafv  41777  aovfundmoveq  41781
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