Image of Relation is Domain 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:
 * $\Img {\mathcal R} = \Dom {\mathcal R^{-1} }$

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

Proof
By definition:
 * $\Img {\mathcal R} := \set {t \in T: \exists s \in S: \tuple {s, t} \in \mathcal R}$
 * $\Dom {\mathcal R^{-1} } := \set {t \in T: \exists s \in S: \tuple {t, s} \in \mathcal R^{-1} }$

Also see

 * Domain of Relation is Image of Inverse Relation