Theorem ndfatafv2 45909

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 45907	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		iffalse 4537	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = 𝒫 ∪ ran 𝐹)
3	1, 2	eqtrid 2784	1 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541  ifcif 4528  𝒫 cpw 4602  ∪ cuni 4908   class class class wbr 5148  ran crn 5677  ℩cio 6493   defAt wdfat 45814  ''''cafv2 45906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-if 4529  df-afv2 45907

This theorem is referenced by:  ndfatafv2undef  45910  ndfatafv2nrn  45919  afv2ndefb  45922  afv20defat  45930