Image of Subset under Mapping is Subset of Image

Corollary of Image of Subset is Subset of Image
Let $S$ and $T$ be sets.

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

Let $A, B \subseteq S$ such that $A \subseteq B$.

Then: the image of $A$ is a subset of the image of $B$:
 * $A \subseteq B \implies f \left({A}\right) \subseteq f \left({B}\right)$

Proof
As $f: S \to T$ is a mapping, it is also a relation, and thus:
 * $f \subseteq S \times T$

The result follows directly from Image of Subset is Subset of Image.