Cartesian Product of Subsets/Family of Subsets

Theorem
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets where $I$ is an arbitrary index set.

Let $S = \displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$.

Let $\family {T_i}_{i \mathop \in I}$ be a family of sets.

Let $T = \displaystyle \prod_{i \mathop \in I} T_i$ be the Cartesian product of $\family {T_i}_{i \mathop \in I}$.

Then:
 * $\paren{\forall i \in I: T_i \subseteq S_i} \implies T \subseteq S$.

Proof
Let $T_i \subseteq S_i$ for all $i \in I$.

Then:

Thus $T \subseteq S$ by the definition of a subset.