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How to prove an almost disjointness preserving operation of $\{0,1\}^n \times \{0,1\}^n$ to $\{0,1\}^n$?
I need help to prove an almost disjointness preserving operation of $\{0,1\}^n \times \{0,1\}^n$ to $\{0,1\}^n$, i.e., an operation satisfying
$f(A)$ is almost disjoint from $B$,
$A \subseteq C$ implies $f(A) \subseteq f(C)$,
for all $A \subseteq \{0,1\}^n \times \{0,1\}^n$ and $B \subseteq \{0,1\}^n$, with $C = \{0,1\}^n$.
(I believe that this is a well-known property, but I was unable to find any reference.)
A:
Here’s a simple argument:
$$
f(A \times B) = f(f^{ -1}(A) \times f^{ -1}(B)) = f(f^{ -1}(A)) \times f(f^{ -1}(B))
$$
and the left-hand side is almost disj
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