Restricting uniformly open surjections

We employ the theory of elementary submodels to improve a recent result by Aron, Jaramillo and Le Donne (Ann. Acad. Sci. Fenn. Math., to appear) concerning restricting uniformly open, continuous surjections to smaller subspaces where they remain surjective. To wit, suppose that $X$ and $Y$ are metric spaces and let $f\colon X\to Y$ be a continuous surjection. If $X$ is complete and $f$ is uniformly open, then $X$ contains a~closed subspace $Z$ with the same density as $Y$ such that $f$ restricted to $Z$ is still uniformly open and surjective. Moreover, if $X$ is a Banach space, then $Z$ may be taken to be a closed linear subspace. A counterpart of this theorem for uniform spaces is also established.