Theorem ndfatafv2nrn

Description: The alternate function value at a class A at which the function is not defined is undefined, i.e., not in the range of the function. (Contributed by AV, 2-Sep-2022)

	Ref	Expression
Assertion	ndfatafv2nrn	⊢ ( ¬ 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		ndfatafv2	⊢ ( ¬ 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) = 𝒫 ∪ ran 𝐹 )
2		pwuninel	⊢ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹
3		df-nel	⊢ ( ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 ↔ ¬ ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 )
4		eleq1	⊢ ( ( 𝐹 '''' 𝐴 ) = 𝒫 ∪ ran 𝐹 → ( ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 ↔ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹 ) )
5	4	notbid	⊢ ( ( 𝐹 '''' 𝐴 ) = 𝒫 ∪ ran 𝐹 → ( ¬ ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹 ) )
6	3 5	syl5bb	⊢ ( ( 𝐹 '''' 𝐴 ) = 𝒫 ∪ ran 𝐹 → ( ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹 ) )
7	2 6	mpbiri	⊢ ( ( 𝐹 '''' 𝐴 ) = 𝒫 ∪ ran 𝐹 → ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 )
8	1 7	syl	⊢ ( ¬ 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 )