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Theorem afvfundmfveq 39867
 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 39861 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
2 iftrue 4042 . 2 (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹𝐴), V) = (𝐹𝐴))
31, 2syl5eq 2656 1 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  Vcvv 3173  ifcif 4036  ‘cfv 5804   defAt wdfat 39842  '''cafv 39843 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-afv 39846 This theorem is referenced by:  afvnufveq  39876  afvfvn0fveq  39879  afv0nbfvbi  39880  afveu  39882  fnbrafvb  39883  afvelrn  39897  afvres  39901  tz6.12-afv  39902  dmfcoafv  39904  afvco2  39905  rlimdmafv  39906  aovfundmoveq  39910
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