Metamath Proof Explorer

Theorem elsymrelsrel

Description: For sets, being an element of the class of symmetric relations ( df-symrels ) is equivalent to satisfying the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021)

		Ref	Expression
	Assertion	elsymrelsrel	$⊢ R \in V \to (R \in SymRels \leftrightarrow SymRel R)$

Proof

Step	Hyp	Ref	Expression
1		elrelsrel	$⊢ R \in V \to (R \in Rels \leftrightarrow Rel R)$
2	1	anbi2d	$⊢ R \in V \to ((R^{-1} \subseteq R \land R \in Rels) \leftrightarrow (R^{-1} \subseteq R \land Rel R))$
3		elsymrels2	$⊢ R \in SymRels \leftrightarrow (R^{-1} \subseteq R \land R \in Rels)$
4		dfsymrel2	$⊢ SymRel R \leftrightarrow (R^{-1} \subseteq R \land Rel R)$
5	2 3 4	3bitr4g	$⊢ R \in V \to (R \in SymRels \leftrightarrow SymRel R)$