Theorem cosscnvssid5

Description: Equivalent expressions for the class of cosets by the converse of the relation R to be a subset of the identity class. (Contributed by Peter Mazsa, 5-Sep-2021)

	Ref	Expression
Assertion	cosscnvssid5	|- ( ( ,~ `' R C_ _I /\ Rel R ) <-> ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) )

Proof

Step	Hyp	Ref	Expression
1		cosscnvssid4	|- ( ,~ `' R C_ _I <-> A. x E* u u R x )
2	1	anbi1i	|- ( ( ,~ `' R C_ _I /\ Rel R ) <-> ( A. x E* u u R x /\ Rel R ) )
3		inecmo3	|- ( ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) <-> ( A. x E* u u R x /\ Rel R ) )
4	2 3	bitr4i	|- ( ( ,~ `' R C_ _I /\ Rel R ) <-> ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) )