Theorem ndfatafv2

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)

	Ref	Expression
Assertion	ndfatafv2	⊢ ( ¬ 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) = 𝒫 ∪ ran 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		df-afv2	⊢ ( 𝐹 '''' 𝐴 ) = if ( 𝐹 defAt 𝐴 , ( ℩ 𝑥 𝐴 𝐹 𝑥 ) , 𝒫 ∪ ran 𝐹 )
2		iffalse	⊢ ( ¬ 𝐹 defAt 𝐴 → if ( 𝐹 defAt 𝐴 , ( ℩ 𝑥 𝐴 𝐹 𝑥 ) , 𝒫 ∪ ran 𝐹 ) = 𝒫 ∪ ran 𝐹 )
3	1 2	syl5eq	⊢ ( ¬ 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) = 𝒫 ∪ ran 𝐹 )