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Metamath Proof Explorer

Theorem ecelqsw

Description: Membership of an equivalence class in a quotient set. More restrictive antecedent; kept for backward compatibility; for new work, prefer ecelqs . (Contributed by Jeff Madsen, 10-Jun-2010) (Revised by Mario Carneiro, 9-Jul-2014) (Proof shortened by AV, 25-Nov-2025)

		Ref	Expression
	Assertion	ecelqsw	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴 ) → [ 𝐵 ] 𝑅 ∈ ( 𝐴 / 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		resexg	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ↾ 𝐴 ) ∈ V )
2		ecelqs	⊢ ( ( ( 𝑅 ↾ 𝐴 ) ∈ V ∧ 𝐵 ∈ 𝐴 ) → [ 𝐵 ] 𝑅 ∈ ( 𝐴 / 𝑅 ) )
3	1 2	sylan	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴 ) → [ 𝐵 ] 𝑅 ∈ ( 𝐴 / 𝑅 ) )