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 e. dom F -> ( F '''' A ) e/ ran F )

Proof

Step	Hyp	Ref	Expression
1		orc	|- ( -. A e. dom F -> ( -. A e. dom F \/ -. Fun ( F |` { A } ) ) )
2		ianor	|- ( -. ( A e. dom F /\ Fun ( F |` { A } ) ) <-> ( -. A e. dom F \/ -. Fun ( F |` { A } ) ) )
3		df-dfat	|- ( F defAt A <-> ( A e. dom F /\ Fun ( F |` { A } ) ) )
4	2 3	xchnxbir	|- ( -. F defAt A <-> ( -. A e. dom F \/ -. Fun ( F |` { A } ) ) )
5	1 4	sylibr	|- ( -. A e. dom F -> -. F defAt A )
6		ndfatafv2nrn	|- ( -. F defAt A -> ( F '''' A ) e/ ran F )
7	5 6	syl	|- ( -. A e. dom F -> ( F '''' A ) e/ ran F )