Let $\left\{ U_{\alpha },\varphi _{\alpha }\right\}$ be a local chart of a regular surface S $\subset \mathbb{R}^{3}$. That is:
$\varphi_{\alpha}:U_{\alpha}\subseteq\mathbb{R}^2\rightarrow \varphi_{\alpha}(U_{\alpha})\subseteq S$ is a $C^\infty$ homeomorphism with injective differential.
We say that a function $f:A\subseteq \mathbb{R} ^{n}\rightarrow \mathbb{R} ^{m} $ is a local diffeomorphism at a point $x\in A$ if there exists a neighborhood $U\subseteq A$ of $x$ such that $f_{|U}:U \rightarrow f(U)$ is bijective, $C^{\infty}$ with $C^{\infty}$ inverse.
Can we conclude that $\varphi_{\alpha}:U_{\alpha}\rightarrow \varphi(U_{\alpha})$ is a local diffeomorphism at every $x\in U_{\alpha}$?
I think it’s true, and I believe that one way to prove it is to apply the inverse function theorem, noting that $d{\varphi_{\alpha}}_{x}:\mathbb{R}^{2}\rightarrow T_{\varphi(x)}S$ is an isomorphism of vector spaces. However, I can’t give a rigorous proof.