Metamath Proof Explorer

Theorem symrelcoss2

Description: The class of cosets by R is symmetric, see dfsymrel2 . (Contributed by Peter Mazsa, 27-Dec-2018)

Ref Expression
Assertion symrelcoss2 
|- ( `' ,~ R C_ ,~ R /\ Rel ,~ R )

Proof

Step Hyp Ref Expression
1 symrelcoss3 
 |-  ( A. x A. y ( x ,~ R y -> y ,~ R x ) /\ Rel ,~ R )
2 cnvsym 
 |-  ( `' ,~ R C_ ,~ R <-> A. x A. y ( x ,~ R y -> y ,~ R x ) )
3 2 anbi1i 
 |-  ( ( `' ,~ R C_ ,~ R /\ Rel ,~ R ) <-> ( A. x A. y ( x ,~ R y -> y ,~ R x ) /\ Rel ,~ R ) )
4 1 3 mpbir 
 |-  ( `' ,~ R C_ ,~ R /\ Rel ,~ R )