Indexed Cartesian Space is Set of all Mappings

Theorem
Let $I$ be an indexing set.

Let $\displaystyle \prod_{i \mathop \in I} S$ denote the cartesian space of $S$ indexed by $I$.

Then $\displaystyle \prod_{i \mathop \in I} S$ is the set of all mappings from $I$ to $S$, and hence the notation:


 * $S^I := \displaystyle \prod_{i \mathop \in I} S$

Proof
Recall the definition of the cartesian space of $S$ indexed by $I$: