Theorem ndfatafv2 46491

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 46489	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		iffalse 4532	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = 𝒫 ∪ ran 𝐹)
3	1, 2	eqtrid 2778	1 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1533  ifcif 4523  𝒫 cpw 4597  ∪ cuni 4902   class class class wbr 5141  ran crn 5670  ℩cio 6487   defAt wdfat 46396  ''''cafv2 46488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-if 4524  df-afv2 46489

This theorem is referenced by:  ndfatafv2undef  46492  ndfatafv2nrn  46501  afv2ndefb  46504  afv20defat  46512