The Stacks project

Proof. Recall that $Z$ is cut out by $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ Y \to g_*\mathcal{O}_ V)$ and $Z'$ is cut out by $\mathcal{I}' = \mathop{\mathrm{Ker}}(\mathcal{O}_ X \to (V \times _ Y X \to X)_*\mathcal{O}_{V \times _ Y X})$, see Lemma 29.6.3. Hence the question is local on $X$ and $Y$ and we may assume $X$ and $Y$ affine. Note that we may replace $V$ by $\coprod V_ i$ where $V = V_1 \cup \ldots \cup V_ n$ is a finite affine open covering. Hence we may assume $g$ is affine. In this case $(V \times _ Y X \to X)_*\mathcal{O}_{V \times _ Y X}$ is the pullback of $g_*\mathcal{O}_ V$ by $f$. Since $f$ is flat we conclude that $f^*\mathcal{I} = \mathcal{I}'$ and the lemma holds. $\square$