Theorem relqmap 38655

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 5775	. 2 ⊢ Rel (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2		df-qmap 38649	. . 3 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
3	2	releqi 5728	. 2 ⊢ (Rel QMap 𝑅 ↔ Rel (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅))
4	1, 3	mpbir 231	1 ⊢ Rel QMap 𝑅

Syntax hints:   ↦ cmpt 5180  dom cdm 5625  Rel wrel 5630  [cec 8635   QMap cqmap 38378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-ss 3919  df-opab 5162  df-mpt 5181  df-xp 5631  df-rel 5632  df-qmap 38649

This theorem is referenced by:  disjqmap2  39029