Let $f:Asubset mathbb{R}^nrightarrow mathbb{R}^n$ an submersion of class $C^1$. Show that $f$ is a local diffeomorphism at each point in open A.

Let $f:Asubset mathbb{R}^nrightarrow mathbb{R}^n$ an submersion of class $C^1$. Show that $f$ is a local diffeomorphism at each point in open A.

I thought so, if $f$ is a submersion then the derivative of $f$ is surjective, consequently we will not have singular points of $f$, that is, we conclude that $det Df(p)neq 0$ for every point $p in A$, it follows that $Df$ is an isomorphism. Right by the Inverse Function Theorem, there are open neighborhoods $V$ and $f(V)$ with $p in V$ and $f(p) in f(V)$ such that $f|_{V}: V rightarrow f(V)$ is a local diffeomorphism of class $C^1$. Is this correct?