Suppose $F:U\rightarrow \mathbb{R}^{n}$ is smooth where $U\subset \mathbb{R}^{m}$ is open with coordinates $(x_{1},...,x_{m})$, and suppose that $a\in U$ is a critical point such that $\frac{\partial F^{1}}{\partial x_{1}}(a)\neq 0$. I want to show that $\phi(x_{1},...,x_{m})=(u,v_{2},...,v_{m})$ where $u=F^{1}(x_{1},..,x_{m})$ and $v_{i}=x_{i}$ for $i=2,..,m$, defines a change of coordinates on some neighborhood of $a$, i.e., is a homeomorphism on some neighborhood of $a$ into $\mathbb{R}^{m}$. This should follow because the corresponding Jacobian is nonsingular at $a$, but what theorem would give me this? How would I write the inverse of $\phi$ in coordinates?
If I knew that $F^{1}$ was injective in some neigherborhood, then I could conclude the same about $\phi$, and hence $\phi$ would be a diffeomorphism. The Jacobian of $F$ is nonsingular at $a$ and so it defines a diffeomorphism in some neighborhood of $a$, but this doesn't imply that $F^{1}$ is injective there.
Edit: What is the coordinate representation of $F$ in these new coordinates?
This is much simpler. The equation $u=F^1(x_1,x_2,..,x_n)$ defines an implicit function $ X_1(u,x_2,...,x_n)$. Just by the Implicit Function Theorem. So the inverse of $\phi$ is just $\psi (u,x_2,..x_n)=(X_1(u,x_2,..x_n),x_2,...x_n)$