I am reading Qing Liu's book on Algebraic Geometry, and on pg. 272, the proof of
lemma 2.20 b) there is a certain part I don't get.
Let $X,Y$ be Noetherian integral schemes and let $f:X \rightarrow Y$ be a separated birational morphism of finite type. Let $ x \in X$ be such that $\mathcal{O}_{Y,y} \rightarrow \mathcal{O}_{X,x}$ where $y=f(x)$ is an isomorphism. I want to show that $$X \times_Y \text{ Spec } \mathcal{O}_{Y,y} \rightarrow \text{Spec } \mathcal{O}_{Y,y}$$ is an isomorphism.
I have no idea how to do this, Liu says one uses an argument similar to the one one uses to show that there is a non-empty open subset $V$ of $Y$ such that $f^{-1}(V) \rightarrow V$ is an isomorphism. To show that, one basically notes that there are non-empty open subsets $U,V$ of $X$ and $Y$ respectively such that $f_U : U \rightarrow V$ is an isomorphism. Then one uses that $f^{-1}(V) \rightarrow V$ is separated and admits a section $V \cong U \subset Y$ to show that $U$ is clopen in $f^{-1}(V).$ Thus, by the irreducility of $f^{-1}(V)$ we have that $f^{-1}(U)=V$ is isomorphic to $V$.
I don't see how to reproduce this proof, or use a similar argument to show the above unfortunately. Could someone help me?