Projection is Injection iff Factor is Singleton/Family of Sets

Theorem
Let $\family {S_i}_{i \mathop \in I}$ be a non-empty family of non-empty 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 $\pr_j: S \to S_j$ be the $j$th projection on $S$.

Then $\pr_j$ is an injection $S_i$ is a singleton for all $i \in I \setminus \set {j}$.