Set Intersection is Self-Distributive/Families of Sets

Theorem
Let $I$ be an indexing set.

Let $\left \langle {A_i} \right \rangle_{i \mathop \in I}$ and $\left \langle {B_i} \right \rangle_{i \mathop \in I}$ be families of subsets of a set $S$.

Then:
 * $\displaystyle \bigcap_{i \mathop \in I} \left({A_i \cap B_i}\right) = \left({\bigcap_{i \mathop \in I} A_i}\right) \cap \left({\bigcap_{i \mathop \in I} B_i}\right)$

where $\displaystyle \bigcap_{i \mathop \in I} A_i$ denotes the intersection of $\left \langle {A_i} \right \rangle$.