One reference is Proposition 14.7 in Remmert's paper *Local Theory of complex analytic spaces*, Several complex variable VII, Encyclopaedia of Math. Sci. vol **74**. For the reader's convenience I will restate the result here.

Recall that a finite, surjective, holomorphic map $\eta \colon X \to Y$ between complex spaces is called a *one sheeted (analytic) covering* if there exists a (critical) thin set $A \subset Y$ such that $\eta^{-1} A$ is thin in $X$ and $\eta \colon X \setminus \eta^{-1}(A) \to Y \setminus A$ is biholomorphic. Then Remmert's statement essentially says that normalizations "dominate" all one-sheeted coverings:

**Proposition.** Let $\eta \colon X \to Y$, $\xi \colon Z \to Y$ be one-sheeted coverings. If $X$ is normal, there exists a unique holomorphic map $g \colon X \to Z$ such that $\eta= \xi \circ g$. If moreover $Z$ is normal, the map $g$ is biholomorphic.

In your setting, take $\eta=f$ and $g=1_Y$. Then if both $X$ and $Y$ are normal Remmert's Proposition tells us that $f$ is a biholomorphism.

`$f_{\ast}(O_X)=O_Y$`

by normality of $Y$, so $f$ is an isomorphism by the link of finiteness and coherent sheaves of algebras). The reduced stalks of`$f_{\ast}(M_X)$`

are products of finite separable extensions of stalk fields of $M_Y$. Since char. 0, write it in "primitive element" form and denominator-chase to get a dense open over which`$f_{\ast}(O_X)$`

is $O_Y$-finite etale of rank $r$. Restrict over $U$ to get $r=1$. $\endgroup$