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 e. V -> ( R e. SymRels <-> SymRel R ) )

Proof

Step	Hyp	Ref	Expression
1		elrelsrel	|- ( R e. V -> ( R e. Rels <-> Rel R ) )
2	1	anbi2d	|- ( R e. V -> ( ( `' R C_ R /\ R e. Rels ) <-> ( `' R C_ R /\ Rel R ) ) )
3		elsymrels2	|- ( R e. SymRels <-> ( `' R C_ R /\ R e. Rels ) )
4		dfsymrel2	|- ( SymRel R <-> ( `' R C_ R /\ Rel R ) )
5	2 3 4	3bitr4g	|- ( R e. V -> ( R e. SymRels <-> SymRel R ) )