Metamath Proof Explorer

Theorem dfsymrels3

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

Ref Expression
Assertion dfsymrels3 
|- SymRels = { r e. Rels | A. x A. y ( x r y -> y r x ) }

Proof

Step Hyp Ref Expression
1 dfsymrels2 
 |-  SymRels = { r e. Rels | `' r C_ r }
2 cnvsym 
 |-  ( `' r C_ r <-> A. x A. y ( x r y -> y r x ) )
3 1 2 rabbieq 
 |-  SymRels = { r e. Rels | A. x A. y ( x r y -> y r x ) }