Metamath Proof Explorer


Theorem eldisjs

Description: Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 24-Jul-2021)

Ref Expression
Assertion eldisjs
|- ( R e. Disjs <-> ( ,~ `' R e. CnvRefRels /\ R e. Rels ) )

Proof

Step Hyp Ref Expression
1 dfdisjs
 |-  Disjs = { r e. Rels | ,~ `' r e. CnvRefRels }
2 cnveq
 |-  ( r = R -> `' r = `' R )
3 2 cosseqd
 |-  ( r = R -> ,~ `' r = ,~ `' R )
4 3 eleq1d
 |-  ( r = R -> ( ,~ `' r e. CnvRefRels <-> ,~ `' R e. CnvRefRels ) )
5 1 4 rabeqel
 |-  ( R e. Disjs <-> ( ,~ `' R e. CnvRefRels /\ R e. Rels ) )