Metamath Proof Explorer

Theorem eqvrelcoss4

Description: Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019) (Revised by Peter Mazsa, 30-Sep-2021)

		Ref	Expression
	Assertion	eqvrelcoss4	\|- ( EqvRel ,~ R <-> A. x A. z ( ( [ x ] ,~ R i^i [ z ] ,~ R ) =/= (/) -> ( [ x ] `' R i^i [ z ] `' R ) =/= (/) ) )

Proof

Step	Hyp	Ref	Expression
1		eqvrelcoss3	\|- ( EqvRel ,~ R <-> A. x A. y A. z ( ( x ,~ R y /\ y ,~ R z ) -> x ,~ R z ) )
2		trcoss2	\|- ( A. x A. y A. z ( ( x ,~ R y /\ y ,~ R z ) -> x ,~ R z ) <-> A. x A. z ( ( [ x ] ,~ R i^i [ z ] ,~ R ) =/= (/) -> ( [ x ] `' R i^i [ z ] `' R ) =/= (/) ) )
3	1 2	bitri	\|- ( EqvRel ,~ R <-> A. x A. z ( ( [ x ] ,~ R i^i [ z ] ,~ R ) =/= (/) -> ( [ x ] `' R i^i [ z ] `' R ) =/= (/) ) )

Step

Hyp

Ref

Expression

eqvrelcoss3

 |-  ( EqvRel ,~ R <-> A. x A. y A. z ( ( x ,~ R y /\ y ,~ R z ) -> x ,~ R z ) )

trcoss2

 |-  ( A. x A. y A. z ( ( x ,~ R y /\ y ,~ R z ) -> x ,~ R z ) <-> A. x A. z ( ( [ x ] ,~ R i^i [ z ] ,~ R ) =/= (/) -> ( [ x ] `' R i^i [ z ] `' R ) =/= (/) ) )

1 2

bitri

 |-  ( EqvRel ,~ R <-> A. x A. z ( ( [ x ] ,~ R i^i [ z ] ,~ R ) =/= (/) -> ( [ x ] `' R i^i [ z ] `' R ) =/= (/) ) )