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Theorem dfatafv2eqfv 46554
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 46552 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), 𝒫 ran 𝐹)
2 iftrue 4530 . 2 (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹𝐴), 𝒫 ran 𝐹) = (𝐹𝐴))
31, 2eqtrid 2779 1 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  ifcif 4524  𝒫 cpw 4598   cuni 4903  ran crn 5673  cfv 6542   defAt wdfat 46409  ''''cafv2 46501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-un 3949  df-if 4525  df-fv 6550  df-afv2 46502
This theorem is referenced by:  afv2rnfveq  46555  afv20fv0  46556  afv2fvn0fveq  46557
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