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Metamath Proof Explorer

Theorem qmapex

Description: Quotient map exists if R exists. Type-safety: ensures QMap is a set under the standard "relation sethood" hypothesis. (Contributed by Peter Mazsa, 12-Feb-2026)

		Ref	Expression
	Assertion	qmapex	⊢ ( 𝑅 ∈ 𝑉 → QMap 𝑅 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		df-qmap	⊢ QMap 𝑅 = ( 𝑥 ∈ dom 𝑅 ↦ [ 𝑥 ] 𝑅 )
2		dmexg	⊢ ( 𝑅 ∈ 𝑉 → dom 𝑅 ∈ V )
3	2	mptexd	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑥 ∈ dom 𝑅 ↦ [ 𝑥 ] 𝑅 ) ∈ V )
4	1 3	eqeltrid	⊢ ( 𝑅 ∈ 𝑉 → QMap 𝑅 ∈ V )