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 ) e/ ran F )

Proof

Step	Hyp	Ref	Expression
1		ndfatafv2	|- ( -. F defAt A -> ( F '''' A ) = ~P U. ran F )
2		pwuninel	|- -. ~P U. ran F e. ran F
3		df-nel	|- ( ( F '''' A ) e/ ran F <-> -. ( F '''' A ) e. ran F )
4		eleq1	|- ( ( F '''' A ) = ~P U. ran F -> ( ( F '''' A ) e. ran F <-> ~P U. ran F e. ran F ) )
5	4	notbid	|- ( ( F '''' A ) = ~P U. ran F -> ( -. ( F '''' A ) e. ran F <-> -. ~P U. ran F e. ran F ) )
6	3 5	syl5bb	|- ( ( F '''' A ) = ~P U. ran F -> ( ( F '''' A ) e/ ran F <-> -. ~P U. ran F e. ran F ) )
7	2 6	mpbiri	|- ( ( F '''' A ) = ~P U. ran F -> ( F '''' A ) e/ ran F )
8	1 7	syl	|- ( -. F defAt A -> ( F '''' A ) e/ ran F )