Theorem dmafv2rnb

Description: The alternate function value at a class A is defined, i.e., in the range of the function, iff A is in the domain of the function. (Contributed by AV, 3-Sep-2022)

	Ref	Expression
Assertion	dmafv2rnb	|- ( Fun ( F |` { A } ) -> ( A e. dom F <-> ( F '''' A ) e. ran F ) )

Proof

Step	Hyp	Ref	Expression
1		iba	|- ( Fun ( F |` { A } ) -> ( A e. dom F <-> ( A e. dom F /\ Fun ( F |` { A } ) ) ) )
2		df-dfat	|- ( F defAt A <-> ( A e. dom F /\ Fun ( F |` { A } ) ) )
3		dfatafv2rnb	|- ( F defAt A <-> ( F '''' A ) e. ran F )
4	2 3	bitr3i	|- ( ( A e. dom F /\ Fun ( F |` { A } ) ) <-> ( F '''' A ) e. ran F )
5	1 4	bitrdi	|- ( Fun ( F |` { A } ) -> ( A e. dom F <-> ( F '''' A ) e. ran F ) )