Metamath Proof Explorer

Theorem elsymrels4

Description: Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021)

Ref Expression
Assertion elsymrels4 
|- ( R e. SymRels <-> ( `' R = R /\ R e. Rels ) )

Proof

Step Hyp Ref Expression
1 dfsymrels4 
 |-  SymRels = { r e. Rels | `' r = r }
2 cnveq 
 |-  ( r = R -> `' r = `' R )
3 id 
 |-  ( r = R -> r = R )
4 2 3 eqeq12d 
 |-  ( r = R -> ( `' r = r <-> `' R = R ) )
5 1 4 rabeqel 
 |-  ( R e. SymRels <-> ( `' R = R /\ R e. Rels ) )