Theorem cnvelrels

Description: The converse of a set is an element of the class of relations. (Contributed by Peter Mazsa, 18-Aug-2019)

	Ref	Expression
Assertion	cnvelrels	$$⊢A\in V\to A^{-1}\in \mathrm{Rels}$$

Proof

Step	Hyp	Ref	Expression
1		relcnv	$$⊢\mathrm{Rel}{A}^{-1}$$
2		cnvexg	$$⊢A\in V\to A^{-1}\in \mathrm{V}$$
3		elrelsrel	$$⊢A^{-1}\in \mathrm{V}\to \left(A^{-1}\in \mathrm{Rels}\leftrightarrow \mathrm{Rel}{A}^{-1}\right)$$
4	2 3	syl	$$⊢A\in V\to \left(A^{-1}\in \mathrm{Rels}\leftrightarrow \mathrm{Rel}{A}^{-1}\right)$$
5	1 4	mpbiri	$$⊢A\in V\to A^{-1}\in \mathrm{Rels}$$