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Theorem afvfundmfveq 41724
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 41718 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
2 iftrue 4236 . 2 (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹𝐴), V) = (𝐹𝐴))
31, 2syl5eq 2806 1 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  Vcvv 3340  ifcif 4230  cfv 6049   defAt wdfat 41699  '''cafv 41700
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 41703
This theorem is referenced by:  afvnufveq  41733  afvfvn0fveq  41736  afv0nbfvbi  41737  afveu  41739  fnbrafvb  41740  afvelrn  41754  afvres  41758  tz6.12-afv  41759  dmfcoafv  41761  afvco2  41762  rlimdmafv  41763  aovfundmoveq  41767
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