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Theorem ndfatafv2 45517
Description: The alternate function value at a class 𝐴 if the function is not defined at this set 𝐴. (Contributed by AV, 2-Sep-2022.)
Assertion
Ref Expression
ndfatafv2 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)

Proof of Theorem ndfatafv2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-afv2 45515 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
2 iffalse 4500 . 2 𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) = 𝒫 ran 𝐹)
31, 2eqtrid 2789 1 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  ifcif 4491  𝒫 cpw 4565   cuni 4870   class class class wbr 5110  ran crn 5639  cio 6451   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-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-if 4492  df-afv2 45515
This theorem is referenced by:  ndfatafv2undef  45518  ndfatafv2nrn  45527  afv2ndefb  45530  afv20defat  45538
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