Projection is Surjection/Family of Sets

Theorem
Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets.

Let $\ds \prod_{\alpha \mathop \in I} S_\alpha$ be the Cartesian product of $\family {S_\alpha}_{\alpha \mathop \in I}$.

Let each of $S_\alpha$ be non-empty.

For each $\beta \in I$, let $\ds \pr_\beta: \prod_{\alpha \mathop \in I} S_\alpha \to S_\beta$ be the $\beta$th projection on $\ds S = \prod_{\alpha \mathop \in I} S_\alpha$.

Then $\pr_\beta$ is a surjection.

Proof
Consider the $\beta$th projection.

Let $x_\beta \in S_\beta$.

Let $\map x \beta = x_\beta$

Suppose $\gamma \in I: \gamma \ne \beta$.

As $S_\gamma \ne \O$ it is possible to use the axiom of choice to choose $\map x \gamma \in S_\gamma$.

Then:
 * $\ds x \in \prod_{\alpha \mathop \in I} S_\alpha$

and:
 * $\map {\pr_\beta} x = \map x \beta$

Hence the result.