Theorem ndfatafv2undef

Description: The alternate function value at a class A is undefined if the function, whose range is a set, is not defined at A . (Contributed by AV, 2-Sep-2022)

	Ref	Expression
Assertion	ndfatafv2undef	⊢ ( ( ran 𝐹 ∈ 𝑉 ∧ ¬ 𝐹 defAt 𝐴 ) → ( 𝐹 '''' 𝐴 ) = ( Undef ‘ ran 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		ndfatafv2	⊢ ( ¬ 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) = 𝒫 ∪ ran 𝐹 )
2		undefval	⊢ ( ran 𝐹 ∈ 𝑉 → ( Undef ‘ ran 𝐹 ) = 𝒫 ∪ ran 𝐹 )
3	2	eqcomd	⊢ ( ran 𝐹 ∈ 𝑉 → 𝒫 ∪ ran 𝐹 = ( Undef ‘ ran 𝐹 ) )
4	1 3	sylan9eqr	⊢ ( ( ran 𝐹 ∈ 𝑉 ∧ ¬ 𝐹 defAt 𝐴 ) → ( 𝐹 '''' 𝐴 ) = ( Undef ‘ ran 𝐹 ) )