The Stacks project

Proof. We will use without further mention that the quasi-compact subsets of $X$ are exactly the closed subsets (Lemma 5.12.5). Consider the collection $\mathcal{E}$ of all quasi-compact subsets $E \subset X$ with $f(E) = Y$ ordered by inclusion. We will use Zorn's lemma to show that $\mathcal{E}$ has a minimal element. To do this it suffices to show that given a totally ordered family $E_\lambda $ of elements of $\mathcal{E}$ the intersection $\bigcap E_\lambda $ is an element of $\mathcal{E}$. It is quasi-compact as it is closed. For every $y \in Y$ the sets $E_\lambda \cap f^{-1}(\{ y\} )$ are nonempty and closed, hence the intersection $\bigcap E_\lambda \cap f^{-1}(\{ y\} ) = \bigcap (E_\lambda \cap f^{-1}(\{ y\} ))$ is nonempty by Lemma 5.12.6. This finishes the proof. $\square$