Let $1\leq \mu <n,\, M\subset \mathbb{R}^n$ be a $\mu$-dimensional differentiable submanifold of $\mathbb{R}^n$ (that is, all the charts $\varphi$ are immersions: $\phi\in C^1$ and the rank of $D\varphi$ equals $\mu$). Now let $f\in C^1(\mathbb{R}^n,\mathbb{R}^n)$ be invertible with $f^{-1}\in C^1(\mathbb{R}^n,\mathbb{R}^n)$.
Show that $f(M)$ is a $\mu$-dimensional submanifold of $\mathbb{R}^n$.
The fact that given $x\in f(M)$ we can find a $C^1$-homeomorphism $\psi:=f\circ\varphi$ follows almost immediately from the assumptions. However, my problem is showing that such a $D(f\circ\varphi)$ has rank $\mu$ and I haven't been able to show this so far.
By the chain rule, $D(f\circ \varphi) = Df\circ D\varphi$. Since $f$ is $C^1$ with $C^1$ inverse, by the chain rule, $D(\mathrm{Identity})= D(f\circ f^{-1}) = Df\circ D(f^{-1})$, so $Df$ is an isomorphism. The rest is a question about linear algebra: if $T$ is an isomorphism, and $S$ is a linear map of rank $\mu$ then $T\circ S$ has rank $\mu$. Apply this to $T = Df$ and $S = D\varphi$.