Proof: Let $\omega\in{\varphi_{\rm trop}}({U^{\rm an}})$. We note that $\varphi_{\rm trop}^{-1}(\omega)$ is a Laurent domain in ${U^{\rm an}}$ and hence it has the same dimension as $U$. We conclude that $\varphi_{\rm trop}^{-1}(\omega)$ is not contained in the analytification of the lower dimensional Zariski-closed subset $U\setminus U^{\prime}$ and hence $\omega\in{\varphi_{\rm trop}}((U^{\prime})^{\rm an})$. $\square$