Theorem riotauni

Description: Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011)

	Ref	Expression
Assertion	riotauni	$$⊢\exists !x\in A\phi \to \left(\iota x\in A|\phi \right)=\bigcup \left\{x\in A|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		df-reu	$$⊢\exists !x\in A\phi \leftrightarrow \exists !x\left(x\in A\wedge \phi \right)$$
2		iotauni	$$⊢\exists !x\left(x\in A\wedge \phi \right)\to \left(\iota x|\left(x\in A\wedge \phi \right)\right)=\bigcup \left\{x|\left(x\in A\wedge \phi \right)\right\}$$
3	1 2	sylbi	$$⊢\exists !x\in A\phi \to \left(\iota x|\left(x\in A\wedge \phi \right)\right)=\bigcup \left\{x|\left(x\in A\wedge \phi \right)\right\}$$
4		df-riota	$$⊢\left(\iota x\in A|\phi \right)=\left(\iota x|\left(x\in A\wedge \phi \right)\right)$$
5		df-rab	$$⊢\left\{x\in A|\phi \right\}=\left\{x|\left(x\in A\wedge \phi \right)\right\}$$
6	5	unieqi	$$⊢\bigcup \left\{x\in A|\phi \right\}=\bigcup \left\{x|\left(x\in A\wedge \phi \right)\right\}$$
7	3 4 6	3eqtr4g	$$⊢\exists !x\in A\phi \to \left(\iota x\in A|\phi \right)=\bigcup \left\{x\in A|\phi \right\}$$