Theorem ndmfvrcl

Description: Reverse closure law for function with the empty set not in its domain (if R = S ). (Contributed by NM, 26-Apr-1996) The class containing the function value does not have to be the domain. (Revised by Zhi Wang, 10-Nov-2025)

	Ref	Expression
Hypotheses	ndmfvrcl.1	$$⊢\mathrm{dom}F=S$$
	ndmfvrcl.2	$$⊢\neg \varnothing \in R$$
Assertion	ndmfvrcl	$$⊢F\left(A\right)\in R\to A\in S$$

Proof

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