Projection is Injection iff Factor is Singleton/Family of Sets/Necessary Condition

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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 $S = \ds \prod_{i \mathop \in I} S_i$ 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 $S$.

Let $\pr_j$ be an injection.


Then $S_i$ is a singleton for all $i \in I \setminus \set j$.


Proof

Let $\pr_j$ be an injection.

Then:

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


We have that $\family {S_i}_{i \mathop \in I}$ is a non-empty family of non-empty sets

Hence, by the axiom of choice (formulation $2$), $S$ is non-empty.

Let $z \in S$.

By the definition of Cartesian product $S$:

$\forall i \in I: \map z i \in S_i$.


Let $x_k \in S_k$ for some $k \in I \setminus \set j$.

Let $z' \in S$ be defined by:

$\map {z'} i = \begin{cases} \map z i & : i \in I \setminus \set k \\ x_k & : i = k \end{cases}$

Then:

\(\ds \map {\pr_j} {z'}\) \(=\) \(\ds \map {z'} j\) Definition of Projection
\(\ds \) \(=\) \(\ds \map z j\) as $j \ne k$
\(\ds \) \(=\) \(\ds \map {\pr_j} z\) Definition of Projection

Thus:

$z' = z$

In particular:

$x_k = \map {z'} k = \map z k = z_k$

It follows that:

$S_k = \set {z_k}$

Since $k$ was arbitrary:

$\forall k \in I \setminus \set j: S_k = \set {z_k}$

The result follows.

$\blacksquare$