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Theorem dfatafv2eqfv 45567
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 45565 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), 𝒫 ran 𝐹)
2 iftrue 4497 . 2 (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹𝐴), 𝒫 ran 𝐹) = (𝐹𝐴))
31, 2eqtrid 2789 1 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  ifcif 4491  𝒫 cpw 4565   cuni 4870  ran crn 5639  cfv 6501   defAt wdfat 45422  ''''cafv2 45514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-un 3920  df-if 4492  df-fv 6509  df-afv2 45515
This theorem is referenced by:  afv2rnfveq  45568  afv20fv0  45569  afv2fvn0fveq  45570
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