Knaster-Tarski Lemma/Corollary/Power Set/Proof 2

Proof
Let $P = \set {x \in \powerset S: x \subseteq \map f x}$.

Let $\ds \bigcup P$ be the union of $P$.

Let $x \in P$.

Then by Set is Subset of Union: General Result:
 * $\ds x \subseteq \bigcup P$

Since $f$ is increasing, $\map f x \subseteq \map f {\bigcup P}$.

By definition of $P$, $x \subseteq \map f x$.

Thus $\ds x \subseteq \map f {\bigcup P}$ by Subset Relation is Transitive:

As this holds for all $x \in P$, $\bigcup P \subseteq \map f {\bigcup P}$ by Union is Smallest Superset: General Result.

Now since $f$ is increasing:


 * $\ds \map f {\bigcup P} \subseteq \map f {\map f {\bigcup P} }$

Thus $\ds \map f {\bigcup P} \in P$ by definition of $P$.

Therefore by Set is Subset of Union: General Result:
 * $\ds \map f {\bigcup P} \subseteq \bigcup P$

We have that:
 * $\ds \bigcup P \subseteq \map f {\bigcup P}$

Thus by definition of set equality:
 * $\ds \map f {\bigcup P} = \bigcup P$

Thus $\ds \bigcup P$ is a fixed point of $f$.