Cartesian Product of Unions/General Result

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

Let $\family {A_i}_{i \mathop \in I}$ and $\family {B_j}_{j \mathop \in J}$ be families of sets indexed by $I$ and $J$ respectively.

Then:
 * $\ds \paren {\bigcup_{i \mathop \in I} A_i} \times \paren {\bigcup_{j \mathop \in J} B_j} = \bigcup_{\tuple {i, j} \mathop \in I \times J} \paren {A_i \times B_j}$

where:
 * $\ds \bigcup_{i \mathop \in I} A_i$ denotes the union of $\family {A_i}_{i \mathop \in I}$ and so on
 * $\times$ denotes Cartesian product.