Theorem rabiun

Description: Abstraction restricted to an indexed union. (Contributed by Brendan Leahy, 26-Oct-2017)

	Ref	Expression
Assertion	rabiun	$$⊢\left\{x\in \bigcup _{y\in A}B|\phi \right\}=\bigcup _{y\in A}\left\{x\in B|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		eliun	$$⊢x\in \bigcup _{y\in A}B\leftrightarrow \exists y\in Ax\in B$$
2	1	anbi1i	$$⊢\left(x\in \bigcup _{y\in A}B\wedge \phi \right)\leftrightarrow \left(\exists y\in Ax\in B\wedge \phi \right)$$
3		r19.41v	$$⊢\exists y\in A\left(x\in B\wedge \phi \right)\leftrightarrow \left(\exists y\in Ax\in B\wedge \phi \right)$$
4	2 3	bitr4i	$$⊢\left(x\in \bigcup _{y\in A}B\wedge \phi \right)\leftrightarrow \exists y\in A\left(x\in B\wedge \phi \right)$$
5	4	abbii	$$⊢\left\{x|\left(x\in \bigcup _{y\in A}B\wedge \phi \right)\right\}=\left\{x|\exists y\in A\left(x\in B\wedge \phi \right)\right\}$$
6		df-rab	$$⊢\left\{x\in \bigcup _{y\in A}B|\phi \right\}=\left\{x|\left(x\in \bigcup _{y\in A}B\wedge \phi \right)\right\}$$
7		iunab	$$⊢\bigcup _{y\in A}\left\{x|\left(x\in B\wedge \phi \right)\right\}=\left\{x|\exists y\in A\left(x\in B\wedge \phi \right)\right\}$$
8	5 6 7	3eqtr4i	$$⊢\left\{x\in \bigcup _{y\in A}B|\phi \right\}=\bigcup _{y\in A}\left\{x|\left(x\in B\wedge \phi \right)\right\}$$
9		df-rab	$$⊢\left\{x\in B|\phi \right\}=\left\{x|\left(x\in B\wedge \phi \right)\right\}$$
10	9	a1i	$$⊢y\in A\to \left\{x\in B|\phi \right\}=\left\{x|\left(x\in B\wedge \phi \right)\right\}$$
11	10	iuneq2i	$$⊢\bigcup _{y\in A}\left\{x\in B|\phi \right\}=\bigcup _{y\in A}\left\{x|\left(x\in B\wedge \phi \right)\right\}$$
12	8 11	eqtr4i	$$⊢\left\{x\in \bigcup _{y\in A}B|\phi \right\}=\bigcup _{y\in A}\left\{x\in B|\phi \right\}$$