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Metamath Proof Explorer

Theorem dfsymrels4

Description: Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 20-Jul-2019)

		Ref	Expression
	Assertion	dfsymrels4	$⊢ SymRels = \{r \in Rels \| r^{-1} = r\}$

Step	Hyp	Ref	Expression
1		dfsymrels2	$⊢ SymRels = \{r \in Rels \| r^{-1} \subseteq r\}$
2		elrelscnveq	$⊢ r \in Rels \to (r^{-1} \subseteq r \leftrightarrow r^{-1} = r)$
3	1 2	rabimbieq	$⊢ SymRels = \{r \in Rels \| r^{-1} = r\}$