I encountered this problem in the FOAG as one step proving the Chevalley’s Theorem (8.4.H of the version April 1, 2023).
Assume that we have proved the fact that “images of closed subsets are constructible under morphisms of finite type between Noetherian schemes”.
Suppose we have a Noetherian ring B, consider the canonical scheme morphism of finite type $ \pi : X=Spec \ B[t] \rightarrow Spec \ B=Y$.
Define the fibral locus for any closed subset $K \subset X$, $ FL(K) $:= { $q \in Y : \pi^{−1} (q) \subset K$ }, and we know that $FL(K)$ is closed.
Now suppose $Z$ is a locally closed subset in $X$, define $ \delta Z := \bar{Z} \setminus Z$ which is also closed as $Z$ is locally closed.
Consider the open subset $Y’ := Y \setminus FL( \bar{Z} )$ and we can restrict $\pi$ to $\pi^{-1}(Y’) \subset X$, call it $\pi’$. Then let $ Z’ := Z \cap \pi^{-1}(Y’)$, similarly we can define $ \delta Z’ := \bar{Z’} \setminus Z’$ where the closure is considered to be the closure in the subspace $\pi^{-1}(Y’)$.
Finally I can state the question. I need to prove that $ \delta Z’$ does not intersect with any generic fiber, i.e the image of it does not contain any generic point in $Y’$.
Here is what I have done. Suppose there is a generic point $ p = \pi’ (q), q \in \delta Z’ $, I have proved that $q = p+(t)$ in the preimage of some distinguished affine open set of $Y’$ (which is also a distinguished affine subscheme of $\pi^{-1}(Y’)$ ) and there is another $ s=p[t] \notin \bar{Z} $, which is also a minimal prime. But I just can’t get any contradiction from what I have deduced. Other approaches are also welcomed.