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:


 * $\Preimg \RR \subseteq S$

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


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

Hence: