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Metamath Proof Explorer

Theorem cosselcnvrefrels5

Description: Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 5-Sep-2021)

Ref Expression
Assertion cosselcnvrefrels5 
|- ( ,~ R e. CnvRefRels <-> ( A. x e. ran R A. y e. ran R ( x = y \/ ( [ x ] `' R i^i [ y ] `' R ) = (/) ) /\ ,~ R e. Rels ) )

Proof

Step Hyp Ref Expression
1 cosselcnvrefrels2 
 |-  ( ,~ R e. CnvRefRels <-> ( ,~ R C_ _I /\ ,~ R e. Rels ) )
2 cossssid5 
 |-  ( ,~ R C_ _I <-> A. x e. ran R A. y e. ran R ( x = y \/ ( [ x ] `' R i^i [ y ] `' R ) = (/) ) )
3 2 anbi1i 
 |-  ( ( ,~ R C_ _I /\ ,~ R e. Rels ) <-> ( A. x e. ran R A. y e. ran R ( x = y \/ ( [ x ] `' R i^i [ y ] `' R ) = (/) ) /\ ,~ R e. Rels ) )
4 1 3 bitri 
 |-  ( ,~ R e. CnvRefRels <-> ( A. x e. ran R A. y e. ran R ( x = y \/ ( [ x ] `' R i^i [ y ] `' R ) = (/) ) /\ ,~ R e. Rels ) )