Preceding implies Image is Subset of Image

Theorem
Let $\struct {S, \precsim}$ be a preordered set.

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

Then $\map \precsim y \subseteq \mathord {\map \precsim x}$

where $\map \precsim y$ denotes the image of $y$ under $\precsim$.

Proof
Let $z \in \mathord {\map \precsim y}$

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

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

Thus by definition of image of element:
 * $z \in \mathord {\map \precsim x}$