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:


 * $\Preimg {\mathcal R} \subseteq S$

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


 * $\Preimg {\mathcal R} = \set {s \in \Dom {\mathcal R}: \exists t \in \Rng {\mathcal R}: \tuple {s, t} \in \mathcal R}$

Hence: