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	$$⊢\mathrm{dom}F=S$$
	ndmfvrcl.2	$$⊢\neg \varnothing \in S$$
Assertion	ndmfvrcl	$$⊢F\left(A\right)\in S\to A\in S$$

Proof

Step	Hyp	Ref	Expression
1		ndmfvrcl.1	$$⊢\mathrm{dom}F=S$$
2		ndmfvrcl.2	$$⊢\neg \varnothing \in S$$
3		ndmfv	$$⊢\neg A\in \mathrm{dom}F\to F\left(A\right)=\varnothing$$
4	3	eleq1d	$$⊢\neg A\in \mathrm{dom}F\to \left(F\left(A\right)\in S\leftrightarrow \varnothing \in S\right)$$
5	2 4	mtbiri	$$⊢\neg A\in \mathrm{dom}F\to \neg F\left(A\right)\in S$$
6	5	con4i	$$⊢F\left(A\right)\in S\to A\in \mathrm{dom}F$$
7	6 1	eleqtrdi	$$⊢F\left(A\right)\in S\to A\in S$$