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Theorem ndfatafv2 47223
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 47221 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
2 iffalse 4534 . 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 1540  ifcif 4525  𝒫 cpw 4600   cuni 4907   class class class wbr 5143  ran crn 5686  cio 6512   defAt wdfat 47128  ''''cafv2 47220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-if 4526  df-afv2 47221
This theorem is referenced by:  ndfatafv2undef  47224  ndfatafv2nrn  47233  afv2ndefb  47236  afv20defat  47244
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