How do you prove the following theorem due to Chevalley? I think a few books(for example EGA) prove this theorem, but it is nice to have a detailed proof of it here. To state the theorem, we need the following definition.
Definition Let $X$ be a topological space. A subset of $X$ is called quasi-constructible if it is a finite union of locally closed subsets.
Noetherian Chevalley's Theorem Let $f\colon X \rightarrow Y$ be a morphism of finite type of schemes. Suppose $Y$ is noetherian. Let $Z$ be a quasi-constructible subset of $X$. Then $f(Z)$ is a quasi-constructible subset of $Y$.
