  We prove that for a finite first order structure $\mathbf{A}$ and a set of
first order formulas $\Phi$ in its language with certain closure properties,
the finitary relations on $A$ that are definable via formulas in $\Phi$ are
uniquely determined by those of arity $|A|^{2}$. This yields new proofs for
some finiteness results from universal algebraic geometry.
