Metamath Proof Explorer


Theorem relqmap

Description: Quotient map is a relation. Guarantees that QMap can be composed, restricted, and used in other relation infrastructure (e.g., membership in Disjs , Rels -based typing). (Contributed by Peter Mazsa, 12-Feb-2026)

Ref Expression
Assertion relqmap Rel QMap 𝑅

Proof

Step Hyp Ref Expression
1 mptrel Rel ( 𝑥 ∈ dom 𝑅 ↦ [ 𝑥 ] 𝑅 )
2 df-qmap QMap 𝑅 = ( 𝑥 ∈ dom 𝑅 ↦ [ 𝑥 ] 𝑅 )
3 2 releqi ( Rel QMap 𝑅 ↔ Rel ( 𝑥 ∈ dom 𝑅 ↦ [ 𝑥 ] 𝑅 ) )
4 1 3 mpbir Rel QMap 𝑅