Image is Subset of Codomain/Corollary 1

Theorem
Let $\mathcal R = S \times T$ be a relation.

The image of $\mathcal R$ is a subset of the codomain of $\mathcal R$:


 * $\operatorname{Im} \left({\mathcal R}\right) \subseteq T$

Corollary
This also holds for mappings:

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

The image of $f$ is a subset of the codomain of $f$:


 * $\operatorname{Im} \left({f}\right) \subseteq T$

Proof of Corollary
As a mapping is by definition also a relation, the result follows immediately.