Theorem brcoss3

Description: Alternate form of the A and B are cosets by R binary relation. (Contributed by Peter Mazsa, 26-Mar-2019)

	Ref	Expression
Assertion	brcoss3	|- ( ( A e. V /\ B e. W ) -> ( A ,~ R B <-> ( [ A ] `' R i^i [ B ] `' R ) =/= (/) ) )

Proof

Step	Hyp	Ref	Expression
1		brcnvg	|- ( ( A e. V /\ u e. _V ) -> ( A `' R u <-> u R A ) )
2	1	elvd	|- ( A e. V -> ( A `' R u <-> u R A ) )
3		brcnvg	|- ( ( B e. W /\ u e. _V ) -> ( B `' R u <-> u R B ) )
4	3	elvd	|- ( B e. W -> ( B `' R u <-> u R B ) )
5	2 4	bi2anan9	|- ( ( A e. V /\ B e. W ) -> ( ( A `' R u /\ B `' R u ) <-> ( u R A /\ u R B ) ) )
6	5	exbidv	|- ( ( A e. V /\ B e. W ) -> ( E. u ( A `' R u /\ B `' R u ) <-> E. u ( u R A /\ u R B ) ) )
7		ecinn0	|- ( ( A e. V /\ B e. W ) -> ( ( [ A ] `' R i^i [ B ] `' R ) =/= (/) <-> E. u ( A `' R u /\ B `' R u ) ) )
8		brcoss	|- ( ( A e. V /\ B e. W ) -> ( A ,~ R B <-> E. u ( u R A /\ u R B ) ) )
9	6 7 8	3bitr4rd	|- ( ( A e. V /\ B e. W ) -> ( A ,~ R B <-> ( [ A ] `' R i^i [ B ] `' R ) =/= (/) ) )