Theorem opelcnv

Description: Ordered-pair membership in converse relation. (Contributed by NM, 13-Aug-1995)

	Ref	Expression
Hypotheses	opelcnv.1	$$⊢A\in \mathrm{V}$$
	opelcnv.2	$$⊢B\in \mathrm{V}$$
Assertion	opelcnv	$$⊢⟨A,B⟩\in R^{-1}\leftrightarrow ⟨B,A⟩\in R$$

Proof

Step	Hyp	Ref	Expression
1		opelcnv.1	$$⊢A\in \mathrm{V}$$
2		opelcnv.2	$$⊢B\in \mathrm{V}$$
3		opelcnvg	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left(⟨A,B⟩\in R^{-1}\leftrightarrow ⟨B,A⟩\in R\right)$$
4	1 2 3	mp2an	$$⊢⟨A,B⟩\in R^{-1}\leftrightarrow ⟨B,A⟩\in R$$