Let $\pi \colon X\to Y$ be a morphism, where $X$ is projective. Suppose the existence of an open subset $U\subset X$ such that $\pi^{-1}(\pi(U))=U$, that $\pi$ is injective on $U$, and that its derivative is injective.

I would like to say that $\pi$ restricts to an isomorphism between $U$ and its image. And I suspect that this should be classical. Does someone knows a proof or a reference for this? (Or a counterexample if it is false..)

Remark: Note that if we omit the assertion $\pi^{-1}(\pi(U))=U$ or $X$ projective, we can easily get counterexamples. We take $\mathbb{P}^1\to C$ the normalisation of a nodal cubic curve $C\subset\mathbb{P}^2$, and let $U\subset \mathbb{P}^1$ be an open subset which is the complement of one point going to the singular point of $C$, then $\pi$ does not restrict to an isomorphism from $U$ to its image (this latter being $C$, which is singular). Taking $X=\mathbb{P}^1$ or $X=U$ shows that both assertions are necessary.