Domain of Relation is Image of Inverse Relation

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

Let $\mathcal R^{-1} \subseteq T \times S$ be the inverse of $\mathcal R$.

Then:
 * $\operatorname{Dom} \left ({\mathcal R}\right) = \operatorname{Im} \left ({\mathcal R^{-1}}\right)$

That is, the domain of a relation is the image of its inverse.

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

Also see

 * Image of Relation is Domain of Inverse Relation