Theorem dfqmap3 39300

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 39304), 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 39298	. 2 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2		df-mpt 5186	. 2 ⊢ (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝑅 ∧ 𝑦 = [𝑥]𝑅)}
3	1, 2	eqtri 2783	1 ⊢ QMap 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝑅 ∧ 𝑦 = [𝑥]𝑅)}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {copab 5166   ↦ cmpt 5185  dom cdm 5647  [cec 8693   QMap cqmap 39027

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-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2752  df-mpt 5186  df-qmap 39298

This theorem is used by:  ecqmap  39301