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Theorem afvfundmfveq 40538
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 40532 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
2 iftrue 4066 . 2 (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹𝐴), V) = (𝐹𝐴))
31, 2syl5eq 2667 1 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  Vcvv 3186  ifcif 4060  cfv 5849   defAt wdfat 40513  '''cafv 40514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-un 3561  df-if 4061  df-fv 5857  df-afv 40517
This theorem is referenced by:  afvnufveq  40547  afvfvn0fveq  40550  afv0nbfvbi  40551  afveu  40553  fnbrafvb  40554  afvelrn  40568  afvres  40572  tz6.12-afv  40573  dmfcoafv  40575  afvco2  40576  rlimdmafv  40577  aovfundmoveq  40581
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