\begin{definition}[Definition:Cartesian Product of Family/Definition 2]
Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets.
The '''Cartesian product of $\family {S_i}_{i \mathop \in I}$''' is the set:
:$\ds \prod_{i \mathop \in I} S_i := \set {f \in \paren {\bigcup_{i \mathop \in I} S_i}^I : \forall i \in I: \paren {\map f i \in S_i} }$
where $\ds \paren {\bigcup_{i \mathop \in I} S_i}^I$ denotes the set of all mappings from $I$ to $\ds \bigcup_{i \mathop \in I} S_i$.
\end{definition}