Let $U$ and $V$ be irreducible varieties and $f\colon V\rightarrow U$ be a proper surjective morphism.
Assume, $f^{-1}(\eta)$ is irreducible ($\eta$ is the generic point of $U$).
$\require{AMScd}$
\begin{CD}
f^{-1}(\eta) @>\displaystyle >> V\\
@V \displaystyle \ V V\ @VV
\displaystyle{f} V\\
\operatorname{Spec}(k(\eta)) @>> \displaystyle > U
\end{CD}
I want to prove
Let $\phi\in H^{0}(V,\mathcal{O}_{V})$, then $\phi \in K(U)$ ($K(U)$ is the function field of $U$).
Intuitively, $\phi\mid_{f^{-1}(\eta)}$ is constant, so $\phi\in k(\eta)$. But, I don't understand how to prove rigorously.
I think I have to consider flat base change,
$\require{AMScd}$
\begin{CD}
f^{-1}(\eta)\otimes \overline{k(\eta)} @>>>f^{-1}(\eta) @>\displaystyle >> V\\
@V \displaystyle \ V V\ @VV
\displaystyle{f\mid_{f^{-1}(\eta)}} V@VV\displaystyle{f} V\\
\operatorname{Spec}(\overline{k(\eta)}) @>>> \operatorname{Spec}(k(\eta)) @>> \displaystyle > U
\end{CD}
I'm stuck. Thanks in advance.
