Theorem relqmap 39352

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 5803	. 2 ⊢ Rel (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2		df-qmap 39346	. . 3 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
3	2	releqi 5754	. 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 5186  dom cdm 5651  Rel wrel 5656  [cec 8699   QMap cqmap 39075

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-ss 3916  df-opab 5168  df-mpt 5187  df-xp 5657  df-rel 5658  df-qmap 39346

This theorem is used by:  disjqmap2  39726