Cartesian Product of Unions/General Result

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

Let $\left\langle{A_i}\right\rangle_{i \mathop \in I}$ and $\left\langle{B_j}\right\rangle_{j \mathop \in J}$ be families of sets indexed by $I$ and $J$ respectively.

Then:
 * $\displaystyle \left({\bigcup_{i \mathop \in I} A_i}\right) \times \left({\bigcup_{j \mathop \in J} B_j}\right) = \bigcup_{\left({i, j}\right) \mathop \in I \times J} \left({A_i \times B_j}\right)$

where:
 * $\displaystyle \bigcup_{i \mathop \in I} A_i$ denotes the union of $\left\langle{A_i}\right\rangle_{i \mathop \in I}$ and so on
 * $\times$ denotes Cartesian product.