Theorem dfsymrels3

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

	Ref	Expression
Assertion	dfsymrels3	$$⊢\mathrm{SymRels}=\left\{r\in \mathrm{Rels}|\forall x\forall y\left(xry\to yrx\right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		dfsymrels2	$$⊢\mathrm{SymRels}=\left\{r\in \mathrm{Rels}|r^{-1}\subseteq r\right\}$$
2		cnvsym	$$⊢r^{-1}\subseteq r\leftrightarrow \forall x\forall y\left(xry\to yrx\right)$$
3	1 2	rabbieq	$$⊢\mathrm{SymRels}=\left\{r\in \mathrm{Rels}|\forall x\forall y\left(xry\to yrx\right)\right\}$$