Let $X$ be a normal projective irreducible surface over an algebraically closed field $k$. Let $\pi\colon Y\to X$ be a birational morphism, such that $Y$ is a smooth projective surface, and assume that no $(-1)$-curve of $Y$ is contracted by $\pi$. How do we prove that there is an isomorphism $\pi^{-1}(X_s)\to X_s$, where $X_s$ denotes the smooth locus of $X$, and that $\pi^{-1}(x)$ is a connected curve for each $x\in X\setminus X_s$?

I would say that it follows from the fact that there exists a unique minimal resolution and that on the smooth locus we only have blow-ups. Does someone has a reference for the statement?