Theorem ndmaovrcl

Description: Reverse closure law, in contrast to ndmovrcl where it is required that the operation's domain doesn't contain the empty set ( -. (/) e. S ), no additional asumption is required. (Contributed by Alexander van der Vekens, 26-May-2017)

	Ref	Expression
Hypothesis	ndmaov.1	$$⊢\mathrm{dom}F=S\times S$$
Assertion	ndmaovrcl	$$⊢((AFB))\in S\to \left(A\in S\wedge B\in S\right)$$

Proof

Step	Hyp	Ref	Expression
1		ndmaov.1	$$⊢\mathrm{dom}F=S\times S$$
2		aovvdm	$$⊢((AFB))\in S\to ⟨A,B⟩\in \mathrm{dom}F$$
3		opelxp	$$⊢⟨A,B⟩\in \left(S\times S\right)\leftrightarrow \left(A\in S\wedge B\in S\right)$$
4	3	biimpi	$$⊢⟨A,B⟩\in \left(S\times S\right)\to \left(A\in S\wedge B\in S\right)$$
5	4 1	eleq2s	$$⊢⟨A,B⟩\in \mathrm{dom}F\to \left(A\in S\wedge B\in S\right)$$
6	2 5	syl	$$⊢((AFB))\in S\to \left(A\in S\wedge B\in S\right)$$