Theorem relqmap 39478

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 5802	. 2 ⊢ Rel (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2		df-qmap 39472	. . 3 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
3	2	releqi 5753	. 2 ⊢ (Rel QMap 𝑅 ↔ Rel (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅))
4	1, 3	mpbir 234	1 ⊢ Rel QMap 𝑅

This proof depends on syntax axioms:   ↦ cmpt 5185  dom cdm 5650  Rel wrel 5655  [cec 8719   QMap cqmap 39201

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-opab 5167  df-mpt 5186  df-xp 5656  df-rel 5657  df-qmap 39472

This theorem is used by:  disjqmap2  39852