Suppose I have a function $f$ on sets. What is the property of $f$ called when, for all sets $x$, $y$: $f(x)$ is a superset of $f(y)$ when $x$ is a superset of $y$i.e.$$\forall x,y : x\supseteq y \Rightarrow f(x) \supseteq f(y)$$
What is the name of the property where $f(A) \supseteq f(B)$ when $A\supseteq B$?
terminology;sets;partial order
It is called monotonicity with respect to the inclusion ordering of sets.More precisely, it is in this case increasing monotonicity since the order is preserved. If the order was reversed, it would be decreasing monotonicity.It can also be called direct monotonicity and reverse monotonicity,  as increasing/decreasing seems used mostly for numeric functions.The function is said to be strictly monotone iff (in the direct case)$$\forall x,y : x\supsetneq y \Rightarrow f(x) \supsetneq f(y)$$