Theorem dfqmap3 39125

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 39129), 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 39123	. 2 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2		df-mpt 5192	. 2 ⊢ (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝑅 ∧ 𝑦 = [𝑥]𝑅)}
3	1, 2	eqtri 2785	1 ⊢ QMap 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝑅 ∧ 𝑦 = [𝑥]𝑅)}

This proof depends on syntax axioms:   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {copab 5172   ↦ cmpt 5191  dom cdm 5660  [cec 8690   QMap cqmap 38852

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-cleq 2754  df-mpt 5192  df-qmap 39123

This theorem is used by:  ecqmap  39126