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Metamath Proof Explorer

Theorem uniqsw

Description: The union of a quotient set. More restrictive antecedent; kept for backward compatibility; for new work, prefer uniqs . (Contributed by NM, 9-Dec-2008) (Proof shortened by AV, 25-Nov-2025)

		Ref	Expression
	Assertion	uniqsw	⊢ ( 𝑅 ∈ 𝑉 → ∪ ( 𝐴 / 𝑅 ) = ( 𝑅 “ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		resexg	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ↾ 𝐴 ) ∈ V )
2		uniqs	⊢ ( ( 𝑅 ↾ 𝐴 ) ∈ V → ∪ ( 𝐴 / 𝑅 ) = ( 𝑅 “ 𝐴 ) )
3	1 2	syl	⊢ ( 𝑅 ∈ 𝑉 → ∪ ( 𝐴 / 𝑅 ) = ( 𝑅 “ 𝐴 ) )