Pre-Image Sigma-Algebra on Domain is Sigma-Algebra

Theorem
Let $X, X'$ be sets, and let $f: X \to X'$ be a mapping.

Let $\mathcal{A}'$ be a $\sigma$-algebra on $X'$.

Then the pre-image $\sigma$-algebra $f^{-1} \left({\mathcal{A}'}\right)$ is a $\sigma$-algebra on $X$.