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 $\mathcal R$:


 * $\forall A \subseteq \Dom {\mathcal R}: \mathcal R \sqbrk A \subseteq T$

In the notation of induced mappings, this can be written as:
 * $\forall A \in \powerset S: \map {\mathcal R^\to} A \in \powerset T$

Corollary 1
These results also hold for mappings: