Directed Suprema Preserving Mapping is Increasing

Theorem
Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice.

Let $f:S \to S$ be a mapping that preserves directed suprema.

Then $f$ is an increasing mapping.

Proof
Let $x, y \in D$ such that
 * $x \preceq y$

Then by definition of reflexivity:
 * $\forall a,b \in \left\{ {x, y}\right\}: \exists z \in \left\{ {x, y}\right\}: a \preceq z \land b \preceq z$

By definition:
 * $\left\{ {x, y}\right\}$ is directed.

By definition of mapping preserves directed suprema:
 * $f$ preserves the supremum of $\left\{ {x, y}\right\}$.

By definition of join semilattice:
 * $\left\{ {x, y}\right\}$ admits a supremum.

By Image of Pair under Mapping:
 * $f\left[{\left\{ {x, y}\right\} }\right] = \left\{ {f\left({x}\right), f\left({y}\right)}\right\}$

Thus by definitions of supremum and upper bound:
 * $f\left({x}\right) \preceq f\left({y}\right)$