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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