Theorem ndfatafv2 47948

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 47946	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		iffalse 4496	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = 𝒫 ∪ ran 𝐹)
3	1, 2	eqtrid 2810	1 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570  ifcif 4487  𝒫 cpw 4562  ∪ cuni 4872   class class class wbr 5109  ran crn 5662  ℩cio 6490   defAt wdfat 47853  ''''cafv2 47945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-if 4488  df-afv2 47946

This theorem is referenced by:  ndfatafv2undef  47949  ndfatafv2nrn  47958  afv2ndefb  47961  afv20defat  47969