Projection is Surjection/Family of Sets

Theorem
Let $\left\langle{S_\alpha}\right\rangle_{\alpha \mathop \in I}$ be a family of sets.

Let $\displaystyle \prod_{\alpha \mathop \in I} S_\alpha$ be the Cartesian product of $\left\langle{S_\alpha}\right\rangle_{\alpha \mathop \in I}$.

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

For each $\beta \in I$, let $\displaystyle \operatorname{pr}_\beta: \prod_{\alpha \mathop \in I} S_\alpha \to S_\beta$ be the $\beta$th projection on $\displaystyle S = \prod_{\alpha \mathop \in I} S_\alpha$

Then $\operatorname{pr}_\beta$ is a surjection.

Proof
Consider the $\beta$th projection.

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

Let $x \left({\beta}\right) = x_\beta$

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

As $S_\gamma \ne \varnothing$ it is possible to use the axiom of choice to choose $x \left({\gamma}\right) \in S_\gamma$.

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

and:
 * $\operatorname{pr}_\beta \left({x}\right) = x \left({\beta}\right)$

Hence the result.