Theorem ndmfvrcl

Description: Reverse closure law for function with the empty set not in its domain. (Contributed by NM, 26-Apr-1996)

	Ref	Expression
Hypotheses	ndmfvrcl.1	|- dom F = S
	ndmfvrcl.2	|- -. (/) e. S
Assertion	ndmfvrcl	|- ( ( F ` A ) e. S -> A e. S )

Proof

Step	Hyp	Ref	Expression
1		ndmfvrcl.1	|- dom F = S
2		ndmfvrcl.2	|- -. (/) e. S
3		ndmfv	|- ( -. A e. dom F -> ( F ` A ) = (/) )
4	3	eleq1d	|- ( -. A e. dom F -> ( ( F ` A ) e. S <-> (/) e. S ) )
5	2 4	mtbiri	|- ( -. A e. dom F -> -. ( F ` A ) e. S )
6	5	con4i	|- ( ( F ` A ) e. S -> A e. dom F )
7	6 1	eleqtrdi	|- ( ( F ` A ) e. S -> A e. S )