Let $F:A→A$ be a function and let $Y⊆A$. Prove that the set $S=\{B:Y⊆B⊆A$ and $F[B]⊆B\}$ is nonempty.
I tried to say that any element $y \in Y$ is in $A$ and we can define the set $B$ such that $y \in B$, but I'm having trouble defining $B$ s.t. $F[B] \subseteq B$ based on the given info.