Metamath Proof Explorer

Theorem dfsymrel5

Description: Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021)

Ref Expression
Assertion dfsymrel5 
|- ( SymRel R <-> ( A. x A. y ( x R y <-> y R x ) /\ Rel R ) )

Proof

Step Hyp Ref Expression
1 dfsymrel2 
 |-  ( SymRel R <-> ( `' R C_ R /\ Rel R ) )
2 relcnveq4 
 |-  ( Rel R -> ( `' R C_ R <-> A. x A. y ( x R y <-> y R x ) ) )
3 2 pm5.32ri 
 |-  ( ( `' R C_ R /\ Rel R ) <-> ( A. x A. y ( x R y <-> y R x ) /\ Rel R ) )
4 1 3 bitri 
 |-  ( SymRel R <-> ( A. x A. y ( x R y <-> y R x ) /\ Rel R ) )