Let $f_1\in C^1(U,\mathbb{R}^n)$ and $f_2\in C^1(V,\mathbb{R}^n)$ be two maps from open connected subsets $U$ and $V$ of $\mathbb{R}^k$ (with $1\leq k \leq n$), such that
- $f_1$ and $f_2$ are injective immersions
- $f_2$ and $f_2$ are homeomorphism onto the same domain $D\subseteq \mathbb{R}^n$
Does this imply that $f_1^{-1}\circ f_2$ is a diffeomorphism from $V$ to $U$ ?
To prove this, I tried to check if the Jacobian of $f_1^{-1}\circ f_2$ is invertible, but I don't really know how to proceed.