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	$$⊢R\in V\to \bigcup \left(A/R\right)=R\left[A\right]$$

Proof

Step	Hyp	Ref	Expression
1		resexg	$$⊢R\in V\to {R\upharpoonright }_A\in \mathrm{V}$$
2		uniqs	$$⊢{R\upharpoonright }_A\in \mathrm{V}\to \bigcup \left(A/R\right)=R\left[A\right]$$
3	1 2	syl	$$⊢R\in V\to \bigcup \left(A/R\right)=R\left[A\right]$$