Theorem dfsymrels5

Description: Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021)

	Ref	Expression
Assertion	dfsymrels5	$$⊢\mathrm{SymRels}=\left\{r\in \mathrm{Rels}|\forall x\forall y\left(xry\leftrightarrow yrx\right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		dfsymrels4	$$⊢\mathrm{SymRels}=\left\{r\in \mathrm{Rels}|r^{-1}=r\right\}$$
2		elrelscnveq2	$$⊢r\in \mathrm{Rels}\to \left(r^{-1}=r\leftrightarrow \forall x\forall y\left(xry\leftrightarrow yrx\right)\right)$$
3	1 2	rabimbieq	$$⊢\mathrm{SymRels}=\left\{r\in \mathrm{Rels}|\forall x\forall y\left(xry\leftrightarrow yrx\right)\right\}$$