Mapping reflects Preordering

Theorem
Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping.

Let ${\precsim} \subseteq T \times T$ be a preordering on $T$.

Let $\mathcal R$ be the relation defined on $S$ by the rule:
 * $x \mathrel {\mathcal R} y \iff \map f x \precsim \map f y$

Then $\mathcal R$ is a preordering on $S$.

Reflexivity
Thus $\mathcal R$ is reflexive.

Transitivity
Thus $\mathcal R$ is transitive.

So, by definition, $\mathcal R$ is a preordering.