Exercise 12, § 1

Let $\Phi$ be a set of mappings of a set A into itself. Let $\mathfrak{F}$ be the subset of $\mathfrak{P}(A)$ consisting of all $X \subset A$ such that $f(X) \subset X$ for each $f \in \Phi$. Show that $\mathfrak{F}$ is a complete lattice with respect to the relation of inclusion.