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:

$$ $$ $$