Let $U \subset \mathbb{R}^n$ be an open and $f: U \rightarrow \mathbb{R}^n$ be a function of class $C^k$ such that $\textrm{det} \ df(x) \neq 0 \ \forall \ x\in U$. Show $f$ is a local diffeomorphism of class $C^k$.
$\textbf{My attempt:}$
By the Inverse Function Theorem, for each $x \in U$ exists $V_x \subset U$ open such that $f(V_x)$ is open, ${f|}_{V_x}: V_x \rightarrow f(V_x)$ is a bijection and ${g|}_{f(V_x)}: f(V_x) \rightarrow V_x$ is its inverse of class $C^k$. Since ${f|}_{V_x}$ is a bijection of class $C^k$ and admits an inverse of class $C^k$, we have that ${f|}_{V_x}$ is a diffeomorphism. Hence $f$ is a local diffeomorphism of class $C^k$.
Is my attempt correct?