Let $R \subseteq U \times V$ be a relation, and $S_0, \dots, S_{m-1} \subseteq U$ Then does the following hold?
$$R\left(\bigcap_{j=0}^{m-1} S_j \right) \subseteq \bigcap_{j=0}^{m-1} R(S_j)$$
It is easily proven that it holds if $R$ is a mapping.
Image is defined as
$R(S) = \{v \in V|\exists u \in S|(u, v) \in R\}$