The Stacks project

Proof. Let $Y' \to Y$ be a surjective étale morphism such that $Y'$ is a disjoint union of affine schemes (Properties of Spaces, Lemma 66.6.1). Let $X' \to X \times _ Y Y'$ be a surjective étale morphism such that $X'$ is a disjoint union of affine schemes. By Lemma 67.30.5 the morphism $X' \to Y'$ is flat. Set $V' = V \times _ Y Y'$. By Lemma 67.16.3 the inverse image of $Z$ in $Y'$ is the scheme theoretic image of $V' \to Y'$ and the inverse image of $Z'$ in $X'$ is the scheme theoretic image of $V' \times _{Y'} X' \to X'$. Since $X' \to X$ is surjective étale, it suffices to prove the result in the case of the morphisms $X' \to Y'$ and $V' \to Y'$. Thus we may assume $X$ and $Y$ are affine schemes. In this case $V$ is a quasi-compact algebraic space. Choose an affine scheme $W$ and a surjective étale morphism $W \to V$ (Properties of Spaces, Lemma 66.6.3). It is clear that the scheme theoretic image of $V \to Y$ agrees with the scheme theoretic image of $W \to Y$ and similarly for $V \times _ Y X \to Y$ and $W \times _ Y X \to X$. Thus we reduce to the case of schemes which is Morphisms, Lemma 29.25.16. $\square$