Theorem ndmafv2nrn

Description: The value of a class outside its domain is not in the range, compare with ndmfv . (Contributed by AV, 2-Sep-2022)

	Ref	Expression
Assertion	ndmafv2nrn	⊢ ( ¬ 𝐴 ∈ dom 𝐹 → ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		orc	⊢ ( ¬ 𝐴 ∈ dom 𝐹 → ( ¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
2		ianor	⊢ ( ¬ ( 𝐴 ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { 𝐴 } ) ) ↔ ( ¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
3		df-dfat	⊢ ( 𝐹 defAt 𝐴 ↔ ( 𝐴 ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
4	2 3	xchnxbir	⊢ ( ¬ 𝐹 defAt 𝐴 ↔ ( ¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
5	1 4	sylibr	⊢ ( ¬ 𝐴 ∈ dom 𝐹 → ¬ 𝐹 defAt 𝐴 )
6		ndfatafv2nrn	⊢ ( ¬ 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 )
7	5 6	syl	⊢ ( ¬ 𝐴 ∈ dom 𝐹 → ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 )