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 $$\mathcal{R}$$:

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