Proof of Knaster-Tarski Theorem

[Fermat1]

Let $F:P(A)->P(A$) be monotone and $C$ be the union of sets whose image is invariant under F. Prove $F(C)=C$

https://i.stack.imgur.com/3Wjdg.png

 

[Evgeny.Makarov]

What is your question?

 

[Fermat1]

What is your question?

Hi, my question is my proof (in the image) correct?

 

[Evgeny.Makarov]

$C$ be the union of sets whose image is invariant under F

So, if I understand correctly, $$C=\bigcup_{X\subseteq A}F(F(X))=F(X)$$. But then it is not clear why $B\in C$ implies $B\in X$ for some $X\subseteq A$ such that $X\subseteq F(X)$.

Why don't you type the proof as text?