Theorem symrelcoss3

Description: The class of cosets by R is symmetric, see dfsymrel3 . (Contributed by Peter Mazsa, 28-Mar-2019) (Revised by Peter Mazsa, 17-Sep-2021)

	Ref	Expression
Assertion	symrelcoss3	|- ( A. x A. y ( x ,~ R y -> y ,~ R x ) /\ Rel ,~ R )

Proof

Step	Hyp	Ref	Expression
1		brcosscnvcoss	|- ( ( x e. _V /\ y e. _V ) -> ( x ,~ R y <-> y ,~ R x ) )
2	1	el2v	|- ( x ,~ R y <-> y ,~ R x )
3	2	biimpi	|- ( x ,~ R y -> y ,~ R x )
4	3	gen2	|- A. x A. y ( x ,~ R y -> y ,~ R x )
5		relcoss	|- Rel ,~ R
6	4 5	pm3.2i	|- ( A. x A. y ( x ,~ R y -> y ,~ R x ) /\ Rel ,~ R )