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Theorem ndfatafv2 42111
 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 42109 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
2 iffalse 4317 . 2 𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) = 𝒫 ran 𝐹)
31, 2syl5eq 2873 1 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1656  ifcif 4308  𝒫 cpw 4380  ∪ cuni 4660   class class class wbr 4875  ran crn 5347  ℩cio 6088   defAt wdfat 42016  ''''cafv2 42108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-if 4309  df-afv2 42109 This theorem is referenced by:  ndfatafv2undef  42112  ndfatafv2nrn  42121  afv2ndefb  42124  afv20defat  42132
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