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
$$ $$ $$

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