Theorem eluniima

Description: Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006)

	Ref	Expression
Assertion	eluniima	$$⊢\mathrm{Fun}F\to \left(B\in \bigcup F\left[A\right]\leftrightarrow \exists x\in AB\in F\left(x\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		funiunfv	$$⊢\mathrm{Fun}F\to \bigcup _{x\in A}F\left(x\right)=\bigcup F\left[A\right]$$
2	1	eleq2d	$$⊢\mathrm{Fun}F\to \left(B\in \bigcup _{x\in A}F\left(x\right)\leftrightarrow B\in \bigcup F\left[A\right]\right)$$
3		eliun	$$⊢B\in \bigcup _{x\in A}F\left(x\right)\leftrightarrow \exists x\in AB\in F\left(x\right)$$
4	2 3	bitr3di	$$⊢\mathrm{Fun}F\to \left(B\in \bigcup F\left[A\right]\leftrightarrow \exists x\in AB\in F\left(x\right)\right)$$