Image is Subset of Codomain

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

For all subsets $$A$$ of the domain $$S$$, the image of $$A$$ is a subset of the range of $$\mathcal{R}$$:


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

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 range of $$f$$:


 * $$\forall A \subseteq \operatorname{Dom} \left ({f}\right): f \left({A}\right) \subseteq \operatorname{Rng} \left ({f}\right)$$

Proof
$$ $$

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