I have a question about an argument in the proof of Proposition 6.2.10 from Liu's "Algebraic Geometry and Arithmetic Curves" (page 223):
Following the proof the problem was already reduced to the case that $f: X \to Y$ is a closed immersion.
A previous result (cor. 2.6) impies that $X$ and $Y$ are smooth over $S$ and have same relative dimension in a neighborhood of $X$ (so the dimension of the fibers coinside locally around $x \in X \subset Y$) with respect the smooth maps $X,Y \to S$.
Why does this alreadt imply that $f:X \to Y$ is an open immersion?