Image of Idempotent and Directed Suprema Preserving Mapping is Complete Lattice

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice.

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

Let $F = \left({f\left[{S}\right], \precsim}\right)$ 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 $f\left[{S}\right]$ such that
 * $D$ admits a supremum in $L$.

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

By definition of ,apping preserves directed suprema:
 * $\sup_L\left({f\left[{D}\right]}\right) = f\left({\sup_L D}\right)$

By definition of idempotent mapping:
 * $f\left[{D}\right] = D$

Thus by definition of image of set:
 * $\sup_L D \in f\left[{S}\right]$

By definition of idempotent mapping:
 * $f\left[{S}\right] = \left\{ {x \in S: f\left({x}\right) = x}\right\}$

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.