Image under Increasing Mapping equal to Special Set is Complete Lattice

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

Let $f:S \to S$ be an increasing mapping.

Let $P = \left({M, \precsim}\right)$ be an ordered subset of $L$ such that
 * $M = \left\{ {x \in S: x = f\left({x}\right)}\right\}$

Then $P$ is complete lattice.