A characterization of Möbius transformations

Let $n\ge 2$ be an integer and let $\mathcal{D}$ be a domain of $\mathbb{R}^n$. Let $f:\mathcal{D}\to\mathbb{R}^n$ be an injective mapping which takes hyperspheres whose interior is contained in $\mathcal{D}$ to hyperspheres in $\mathbb{R}^n$. Then $f$ is the restriction of a Möbius transformation.