Intersection Distributes over Union/Family of Sets/Corollary

Theorem
Let $I$ and $J$ be indexing sets.

Let $\left \langle {A_\alpha} \right \rangle_{\alpha \mathop \in I}$ and $\left \langle {B_\beta} \right \rangle_{\beta \mathop \in J}$ be indexed families of subsets of a set $S$.

Then:
 * $\displaystyle \bigcup_{\left({\alpha, \beta}\right) \mathop \in I \times J} \left({A_\alpha \cap B_\beta}\right) = \left({\bigcup_{\alpha \mathop \in I} A_\alpha}\right) \cap \left({\bigcup_{\beta \mathop \in J} B_\beta}\right)$

where $\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\left \langle {A_\alpha} \right \rangle_{\alpha \mathop \in I}$.