Preceding implies Image is Subset of Image

Theorem
Let $\left({S, \precsim}\right)$ be a preordered set.

Let $x, y \in S$ such that
 * $x \precsim y$

Then $\precsim\left({y}\right) \subseteq \mathord\precsim\left({x}\right)$

where $\precsim\left({y}\right)$ denotes the image of $y$ under $\precsim$.

Proof
Let $z \in \mathord\precsim\left({y}\right)$

By definition of image of element:
 * $y \precsim z$

By definition of transitivity:
 * $x \precsim z$

Thus by definition of image of element:
 * $z \in \mathord\precsim\left({x}\right)$