Theorem relqmap 39082

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.)

Assertion

Ref	Expression
relqmap	⊢ Rel QMap 𝑅

Proof of Theorem relqmap

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mptrel 5814	. 2 ⊢ Rel (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2		df-qmap 39076	. . 3 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
3	2	releqi 5766	. 2 ⊢ (Rel QMap 𝑅 ↔ Rel (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅))
4	1, 3	mpbir 234	1 ⊢ Rel QMap 𝑅

Syntax hints:   ↦ cmpt 5193  dom cdm 5663  Rel wrel 5668  [cec 8693   QMap cqmap 38805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-ss 3923  df-opab 5175  df-mpt 5194  df-xp 5669  df-rel 5670  df-qmap 39076

This theorem is referenced by:  disjqmap2  39456