Theorem dfqmap3 38830

Description: Alternate definition of the quotient map: QMap as ordered-pair class abstraction. Gives the raw set-builder characterization for extensional proofs, Rel proofs (relqmap 38834), and composition/intersection manipulations. (Contributed by Peter Mazsa, 14-Feb-2026.)

Assertion

Ref	Expression
dfqmap3	⊢ QMap 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝑅 ∧ 𝑦 = [𝑥]𝑅)}

Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem dfqmap3

Step	Hyp	Ref	Expression
1		df-qmap 38828	. 2 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2		df-mpt 5157	. 2 ⊢ (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝑅 ∧ 𝑦 = [𝑥]𝑅)}
3	1, 2	eqtri 2764	1 ⊢ QMap 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝑅 ∧ 𝑦 = [𝑥]𝑅)}

Syntax hints:   ∧ wa 397   = wceq 1548   ∈ wcel 2121  {copab 5137   ↦ cmpt 5156  dom cdm 5621  [cec 8635   QMap cqmap 38557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-cleq 2733  df-mpt 5157  df-qmap 38828

This theorem is referenced by:  ecqmap  38831