Greetings,

(I suspect this question has nothing to do with the Valuation Criterion of Properness, but I don't know for sure - feel free to modify my title)

This question arises in section 2.4 of Fulton's book on Toric Varieties - in the proof that $\phi_* : X(\Delta') \to X(\Delta)$ is proper iff $\phi^{-1}(|\Delta|)=|\Delta'|$.

Let's say I give you a variety map $f: X \to Y$. To prove it's proper I must take any dvr R with fraction field $K$ and any commutative diagram where one path is $Spec(K) \to Spec(R) \to Y$ and another path is $Spec(K) \to X \to Y$ (the map $X \to Y$ is $f$), and tell you why there exists a unique map $Spec(R) \to X$ that makes both 'triangles' commute. (I don't know how to make pretty diagrams on math overflow, apologies.)

The claim that is made is: Let $U \subseteq X$ be your favorite open subset. If $X$ is irreducible, then to prove the above we may assume that $im(Spec(K) \to X) \subseteq U$.

Will someone please explain why this claim holds?

Robert