Theorem bj-dfrel4v

Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 for relations. (Contributed by Mario Carneiro, 16-Aug-2015) Do not depend on df-cnv , reverse equality. (Revised by BJ, 16-Oct-2026) Note: this theorem could be used in place of opabid2 , which could be deleted.

	Ref	Expression
Assertion	bj-dfrel4v	$$⊢\mathrm{Rel}R\leftrightarrow \left\{⟨x,y⟩|xRy\right\}=R$$

Proof

Step	Hyp	Ref	Expression
1		bj-dfrel3	$$⊢\mathrm{Rel}R\leftrightarrow {R\upharpoonright }_{\mathrm{V}}=R$$
2		relpart	$$⊢{R\upharpoonright }_{\mathrm{V}}=\left\{⟨x,y⟩|xRy\right\}$$
3	2	eqeq1i	$$⊢{R\upharpoonright }_{\mathrm{V}}=R\leftrightarrow \left\{⟨x,y⟩|xRy\right\}=R$$
4	1 3	bitri	$$⊢\mathrm{Rel}R\leftrightarrow \left\{⟨x,y⟩|xRy\right\}=R$$