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 range of $$\mathcal{R}$$:


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

Corollary
This also holds for mappings:

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

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


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

Proof
$$ $$ $$

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