Given a function from a set $X$ to a set $Y$ such then I am trying to prove or disprove that if $f|\_U$ is a bijection then for any $V\in\mathcal P(X)$ not disjoint from $U$ the equality
$$
f[V\cap U]=f[V]\cap f[U]
$$
holds. So by a general theorem I know that
$$
f[V\cap U]\subseteq f[V]\cap f[U]
$$
but unfortunately I am not able to prove or disprove the reverse inclusion: so could someone help me, please?