How to prove $f^{-1}(V) \cong f^{-1}(U) \times_U V$, where $f: X \to Y$ is a morphism of schemes

Question

Let $f: X \to Y$ be a morphism of schemes. Let $U$ be an affine open subset of $Y$, and $V \subseteq U$ an open subset of $Y$. I am trying to prove $$ f^{-1}(V) \cong f^{-1}(U) \times_U V. $$
I thought it's just a diagram chasing but it is not working out.. Any explanation is appreciated. Thank you.