In this paper, we show various hardness results for the satisfiability of ground sets of literals in the theory of equality and uninterpreted functions (EUF) with finite cardinality constraints. It is known that this problem is NP-complete in the general case, when a domain D has a finite cardinality constraint |D| ≤ n, for any n ≥ 3. We notice that this problem stays NP-complete in the case of a single domain of cardinality 2, indicating that even the Boolean sort can be hard to handle when functions are present. It is trivial that the function-free case with Boolean sorts has good complexity, but allowing a single binary function, or (an arbitrary number of) unary functions only, makes the problem NP-complete.