Theorem ndfatafv2 44703

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.

Step	Hyp	Ref	Expression
1		df-afv2 44701	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		iffalse 4468	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = 𝒫 ∪ ran 𝐹)
3	1, 2	eqtrid 2790	1 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539  ifcif 4459  𝒫 cpw 4533  ∪ cuni 4839   class class class wbr 5074  ran crn 5590  ℩cio 6389   defAt wdfat 44608  ''''cafv2 44700

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-if 4460  df-afv2 44701

This theorem is referenced by:  ndfatafv2undef  44704  ndfatafv2nrn  44713  afv2ndefb  44716  afv20defat  44724