Theorem cnvcnv3

Description: The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015)

	Ref	Expression
Assertion	cnvcnv3	$$⊢{R^{-1}}^{-1}=\left\{⟨x,y⟩|xRy\right\}$$

Proof

Step	Hyp	Ref	Expression
1		df-cnv	$$⊢{R^{-1}}^{-1}=\left\{⟨x,y⟩|y{R}^{-1}x\right\}$$
2		vex	$$⊢y\in \mathrm{V}$$
3		vex	$$⊢x\in \mathrm{V}$$
4	2 3	brcnv	$$⊢y{R}^{-1}x\leftrightarrow xRy$$
5	4	opabbii	$$⊢\left\{⟨x,y⟩|y{R}^{-1}x\right\}=\left\{⟨x,y⟩|xRy\right\}$$
6	1 5	eqtri	$$⊢{R^{-1}}^{-1}=\left\{⟨x,y⟩|xRy\right\}$$