Image of Idempotent and Directed Suprema Preserving Mapping is Complete Lattice

Theorem
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.

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

Let $F = \struct {f \sqbrk S, \precsim}$ be an ordered subset of $L$.

Then $F$ inherits directed suprema and is complete lattice.

Proof
We will prove that
 * $F$ inherits directed suprema.

Let $D$ be a directed subset of $f \sqbrk S$ such that
 * $D$ admits a supremum in $L$.

By definition of ordered subset:
 * $D$ is directed in $L$.

By definition of mapping preserves directed suprema:
 * $\map {\sup_L} {f \sqbrk D} = \map f {\sup_L D}$

By definition of idempotent mapping:
 * $f \sqbrk D = D$

Thus by definition of image of set:
 * $\sup_L D \in f \sqbrk S$

By definition of idempotent mapping:
 * $f \sqbrk S = \set {x \in S: \map f x = x}$

By Directed Suprema Preserving Mapping is Increasing:
 * $f$ is an increasing mapping.

Thus by Image under Increasing Mapping equal to Special Set is Complete Lattice:
 * $F$ is complete lattice.