Meet-Continuous implies Shift Mapping Preserves Directed Suprema

Theorem
Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be a meet-continuous lattice.

Let $x \in S$.

Let $f: S \to S$ be a mapping such that:
 * $\forall y \in S: \map f y = x \wedge y$

Then:
 * $f$ preserves directed suprema.

Proof
Let $D$ be a directed subset of $S$ such that
 * $D$ admits a supremum.

By Singleton is Directed and Filtered Subset:
 * $\set x$ is directed.

By Up-Complete Product/Lemma 1:
 * $\set x \times D$ is directed in the simple order product $\struct {S \times S, \precsim}$ of $\mathscr S$ and $\mathscr S$.

Define a mapping $g: S \times S \to S$:
 * $\forall s, t \in S: \map g {s, t} = s \wedge t$

By Meet-Continuous iff Meet Preserves Directed Suprema:
 * $g$ preserves directed suprema.

By definition of meet-continuous:
 * $\mathscr S$ is up-complete.

By Up-Complete Product:
 * $\struct {S \times S, \precsim}$ is up-complete.

By definition of up-complete:
 * $\set x \times D$ admits a supremum.

Thus