Preimage of Relation is Subset of Domain

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

Then the preimage of $\mathcal R$ is a subset of its domain:


 * $\operatorname{Im}^{-1} \left({\mathcal R}\right) \subseteq S$

Proof
The preimage of $\mathcal R$ is defined as:


 * $\operatorname{Im}^{-1} \left ({\mathcal R}\right) = \left\{{s \in \operatorname{Dom} \left({\mathcal R}\right): \exists t \in \operatorname{Rng} \left({\mathcal R}\right): \left({s, t}\right) \in \mathcal R}\right\}$

Hence: