MHB An increasing function on the power set of a set

[caffeinemachine]

Let $A$ be a finite set. Let $\mathcal{P}(A)$ denote the power set of $A$. Let $f:\mathcal{P}(A)\to \mathcal{P}(A)$ be a function such that $X\subseteq Y\Rightarrow f(X)\subseteq f(Y)$. Show that $\exists T\in \mathcal{P}(A)$ such that $f(T)=T$.

P.S. The power set of $A$ is the set of all the subsets of $A$.
NOTE: The theorem holds even when $A$ is not finite.

 

[Evgeny.Makarov]

This is a special case of Knaster–Tarski theorem.

 

[caffeinemachine]

This is a special case of Knaster–Tarski theorem.

I didn't know about this theorem. Thanks.