Preimage of Serial Relation is Domain

Theorem
Let $\mathcal R$ be a serial relation on $S$.

Then the preimage of $\mathcal R$ is $S$ (the domain of $\mathcal R$).

Proof
Hence the result:
 * $\operatorname{Im}^{-1} \left ({\mathcal R}\right) = \operatorname{Dom} \left ({\mathcal R}\right)$