Theorem ndfatafv2 48554

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 48552	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		iffalse 4494	. 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)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570  ifcif 4485  𝒫 cpw 4560  ∪ cuni 4872   class class class wbr 5108  ran crn 5659  ℩cio 6518   defAt wdfat 48460  ''''cafv2 48551

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-if 4486  df-afv2 48552

This theorem is used by:  ndfatafv2undef  48555  ndfatafv2nrn  48564  afv2ndefb  48567  afv20defat  48575