Let $(X,\mathcal{X})$ and $(Y,\mathcal{Y})$ be measurable spaces.
We consider the product of the two spaces, denoted $(X \times Y,\mathcal{X} \overline{\times} \mathcal{Y})$, where $\mathcal{X} \overline{\times} \mathcal{Y}$ is the smallest $\sigma$-algebra generated by $\{ A \times B \mid A \in \mathcal{X},B \in \mathcal{Y} \}$.
For a measurable set $S \in \mathcal{X} \overline{\times} \mathcal{Y}$ and an element $x \in X$, wikipedia states that the following set is measurable: $$ S_x := \{ y \in Y \mid (x,y) \in S \} $$
How can I prove this?