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Metamath Proof Explorer

Theorem dfqmap3

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 ), and composition/intersection manipulations. (Contributed by Peter Mazsa, 14-Feb-2026)

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

Proof

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