Theorem dfqmap3 39047

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 39051), 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 39045	. 2 ⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)
2		df-mpt 5198	. 2 ⊢ (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝑅 ∧ 𝑦 = [𝑥]𝑅)}
3	1, 2	eqtri 2793	1 ⊢ QMap 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝑅 ∧ 𝑦 = [𝑥]𝑅)}

Syntax hints:   ∧ wa 400   = wceq 1568   ∈ wcel 2150  {copab 5178   ↦ cmpt 5197  dom cdm 5665  [cec 8695   QMap cqmap 38774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-cleq 2762  df-mpt 5198  df-qmap 39045

This theorem is referenced by:  ecqmap  39048