Image is Subset of Codomain

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

For all subsets $$A$$ of the domain of $$\mathcal R$$, the image of $$A$$ is a subset of the codomain of $$\mathcal R$$:


 * $$\forall A \subseteq \operatorname{Dom} \left ({\mathcal R}\right): \mathcal R \left({A}\right) \subseteq T$$

In the language of induced mappings, this can be written as:
 * $$\forall A \in \mathcal P \left({S}\right): f_{\mathcal R} \left({A}\right) \in \mathcal P \left({T}\right)$$

Corollary
This also holds for mappings:

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

For all subsets $$A$$ of the domain $$S$$, the image of $$A$$ is a subset of the codomain of $$f$$:


 * $$\forall A \subseteq S: f \left({A}\right) \subseteq T$$

Proof
$$ $$ $$

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