Directed Suprema Preserving Mapping is Increasing

Theorem
Let $L = \struct {S, \vee, \preceq}$ 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:
 * $\forall a, b \in \set {x, y}: \exists z \in \set {x, y}: a \preceq z \land b \preceq z$

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

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

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

By Image of Doubleton under Mapping:
 * $f \sqbrk {\set {x, y} } = \set {\map f x, \map f y}$

Thus by definitions of supremum and upper bound:
 * $\map f x \preceq \map f y$