The range of holomorphic maps at boundary points

We prove a boundary version of the open mapping theorem for holomorphic maps between strongly pseudoconvex domains. That is, we prove that the local image of a holomorphic map $f:D\to D^{\mathrm{\prime }}$ close to a boundary regular contact point $p\in \text{de}D$ where the Jacobian is bounded from zero along normal non-tangential directions has to eventually contain every cone (and more generally every admissible region) with vertex at $f(p)$.