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Metamath Proof Explorer

Theorem eldisjsim4

Description: Disjs implies element-disjoint range of QMap . Same as eldisjsim3 but expressed using the block-map range ran QMap R (often the more modular expression). (Contributed by Peter Mazsa, 15-Feb-2026)

		Ref	Expression
	Assertion	eldisjsim4	⊢ ( 𝑅 ∈ Disjs → ran QMap 𝑅 ∈ ElDisjs )

Proof

Step	Hyp	Ref	Expression
1		rnqmap	⊢ ran QMap 𝑅 = ( dom 𝑅 / 𝑅 )
2		eldisjsim3	⊢ ( 𝑅 ∈ Disjs → ( dom 𝑅 / 𝑅 ) ∈ ElDisjs )
3	1 2	eqeltrid	⊢ ( 𝑅 ∈ Disjs → ran QMap 𝑅 ∈ ElDisjs )