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Theorem dfatafv2eqfv 46700
Description: If a function is defined at a class 𝐴, the alternate function value equals the function's value at 𝐴. (Contributed by AV, 3-Sep-2022.)
Assertion
Ref Expression
dfatafv2eqfv (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹𝐴))

Proof of Theorem dfatafv2eqfv
StepHypRef Expression
1 dfafv22 46698 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), 𝒫 ran 𝐹)
2 iftrue 4531 . 2 (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹𝐴), 𝒫 ran 𝐹) = (𝐹𝐴))
31, 2eqtrid 2777 1 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  ifcif 4525  𝒫 cpw 4599   cuni 4904  ran crn 5674  cfv 6543   defAt wdfat 46555  ''''cafv2 46647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-un 3946  df-if 4526  df-fv 6551  df-afv2 46648
This theorem is referenced by:  afv2rnfveq  46701  afv20fv0  46702  afv2fvn0fveq  46703
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