I would like to understand and prove the following two "well-known" facts:
If $B$ is a scheme and $P$ a property for which I know:
i) if $B=Spec(V)$ where $V$ is a complete DVR, if $P$ holds on the special point then it holds on the generic point of $B$
ii) $P$ is true on a constructible set of $B$
then $P$ holds on an open subset of $B$.
Assume $B$ integral. If $P$ true on the generic fiber implies that there exists an open dense of $B$ where $P$ holds then $P$ holds on a constructible set.