I wish to know if a theorem like the following exists, or if there are any obvious counterexamples. This theorem generalizes Tarski's fixed point theorem in the case of powersets.
Let $X,Y$ be non-empty sets and $2^X$ the powerset of $X$.
Let $f : 2^X \times Y \to 2^X$ be such that for all $x,x' \in 2^X, y \in Y$, if $x \subseteq x'$, then $f(x,y) \subseteq f(x',y)$.
Then $f$ has at least one fixed point: there exists some $x \in 2^X$ such that for all $y \in Y, x = f(x,y)$.