Projection is Injection iff Factor is Singleton/Family of Sets

Theorem
Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets where $I$ is the indexing 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 $i \in I \setminus \set {j}$.

Sufficient Condition
Suppose $S_k = \set {s_k}$ for all $k \in \set {1, 2, \dotsc, n}$ where $k \ne j$.

Let $\map {\pr_j} x = \map {\pr_j} y = z$ for $x, y \in S$.

Then by definition of $j$th projection, $x, y \in S$ are given by:
 * $x = \tuple {s_1, s_2, \dotsc, s_{j - 1}, z, s_{j + 1}, \dotsc, s_n}$
 * $y = \tuple {s_1, s_2, \dotsc, s_{j - 1}, z, s_{j + 1}, \dotsc, s_n}$

and so $x = y$.

Thus $\pr_j$ is an injection by definition.

Necessary Condition
Let $\pr_j$ be an injection.

Thus:
 * $\forall x, y \in S: \map {\pr_j} x = \map {\pr_j} y \implies x = y$

Let $x, y \in S$ be such that:
 * $x = \tuple {x_1, x_2, \dotsc, x_{j - 1}, z, x_{j + 1}, \dotsc, x_n}$
 * $y = \tuple {y_1, y_2, \dotsc, y_{j - 1}, z, y_{j + 1}, \dotsc, y_n}$

Then:
 * $\map {\pr_j} x = \map {\pr_j} y = z$

For $x = y$ it is necessary that:
 * $x_1 = y_1, x_2 = y_2, \dotsc, x_{j - 1} = y_{j - 1}, x_{j + 1} = y_{j + 1}, \dotsc, x_n = y_n$

That is:
 * $\forall k \in \set {1, 2, \dotsc, n}, k \ne j: x_k = y_k$

That is, that $S_k$ is a singleton for all $k \in \set {1, 2, \dotsc, n}$ where $k \ne j$.