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