Theorem imauni

Description: The image of a union is the indexed union of the images. Theorem 3K(a) of Enderton p. 50. (Contributed by NM, 9-Aug-2004) (Proof shortened by Mario Carneiro, 18-Jun-2014)

	Ref	Expression
Assertion	imauni	$$⊢A\left[\bigcup B\right]=\bigcup _{x\in B}A\left[x\right]$$

Proof

Step	Hyp	Ref	Expression
1		uniiun	$$⊢\bigcup B=\bigcup _{x\in B}x$$
2	1	imaeq2i	$$⊢A\left[\bigcup B\right]=A\left[\bigcup _{x\in B}x\right]$$
3		imaiun	$$⊢A\left[\bigcup _{x\in B}x\right]=\bigcup _{x\in B}A\left[x\right]$$
4	2 3	eqtri	$$⊢A\left[\bigcup B\right]=\bigcup _{x\in B}A\left[x\right]$$