Let $f\colon A \longrightarrow B$ be a function, and let $U \subset A$ be a subset. The direct image of $U$ is the set $f(U) \subset B$ consisting of all elements of $B$ which equal $f(u)$ for some $u \in U$

Direct images satisfy the following properties:

1. Unions: For any collection $\{U_i\}_{i \in I}$ of subsets of $A$ $$ f\left(\bigcup_{i \in I} U_i\right) = \bigcup_{i\in I} f(U_i). $$
2. Intersections: For any collection $\{U_i\}_{i \in I}$ of subsets of $A$ $$ f\left(\bigcap_{i \in I} U_i\right) \subset \bigcap_{i\in I} f(U_i). $$
3. Set difference: For any $U,V \subset A$ $$ f(V \setminus U) \supset f(V) \setminus f(U). $$ In particular, the complement of $U$ satisfies $f(U^\complement) \supset f(A) \setminus f(U)$
4. Subsets: If $U \subset V \subset A$ then $f(U) \subset f(V) \subset B$
5. Inverse image of a direct image: For any $U \subset A$ $$f^{-1}(f(U)) \supset U$$ with equality if $f$ is injective.
6. Direct image of an inverse image: For any $V \subset B$ $$f(f^{-1}(V)) \subset V$$ with equality if $f$ is surjective.