Metamath Proof Explorer


Theorem dfcoss3

Description: Alternate definition of the class of cosets by R (see the comment of df-coss ). (Contributed by Peter Mazsa, 27-Dec-2018)

Ref Expression
Assertion dfcoss3 R = R R -1

Proof

Step Hyp Ref Expression
1 brcnvg x V u V x R -1 u u R x
2 1 el2v x R -1 u u R x
3 2 anbi1i x R -1 u u R y u R x u R y
4 3 exbii u x R -1 u u R y u u R x u R y
5 4 opabbii x y | u x R -1 u u R y = x y | u u R x u R y
6 df-co R R -1 = x y | u x R -1 u u R y
7 df-coss R = x y | u u R x u R y
8 5 6 7 3eqtr4ri R = R R -1