downward Lowenheim-Skolem theorem

Let $L$ be a first order language, let $𝒜$ be an $L$-structure and let $K\subseteq \mathrm{dom}(𝒜)$. Then there is an $L$-structure $ℬ$ such that $K\subseteq ℬ$ and $|ℬ|\le \mathrm{Max}(|K|,|L|)$ and $ℬ$ is elementarily embedded in $𝒜$.