Image Preserves Subsets

Theorem
Let $f: S \to T$ be a mapping.

Let $S'$ and $S$ be sets such that $S' \subseteq S \subseteq S$.

Then, the image of $S'$ under $f$ is a subset of the image of $S''$ under $f$.

Proof
Suppose that $t \in \operatorname{Im} \left({S'}\right)$, i.e., for some $s \in S'$:


 * $f \left({s}\right) = t$

Then since $S' \subseteq S''$, we also have that:


 * $s \in S''$

by definition of subset.

Hence by definition of image:


 * $t \in \operatorname{Im} \left({S''}\right)$

The result follows, by definition of subset.