The Stacks project

Proof. We may assume that $Y = \mathop{\mathrm{Spec}}(R)$ and $f \in R$. Consider the morphism of affine schemes $\phi : U = \mathop{\mathrm{Spec}}(R_ f) \to \mathop{\mathrm{Spec}}(R) = Y$ induced by the ring map $R \to R_ f$. By Algebra, Lemma 10.17.6 we know that it is a homeomorphism onto $D(f)$. On the other hand, the map $\phi ^{-1}\mathcal{O}_ Y \to \mathcal{O}_ U$ is an isomorphism on stalks, hence an isomorphism. Thus we see that $\phi $ is an open immersion. We conclude that $D(f)$ is isomorphic to $U$ by Lemma 26.3.4. $\square$