Given an uncountable regular cardinal $\kappa$, we study the structural properties of the class of all sets of functions from $\kappa$ to $\kappa$ that are definable over the structure $\langle \mathrm{H}({\kappa }^{+}),\in \rangle$ by a ${\Sigma }_1$-formula with parameters. It is well known that many important statements about these classes are not decided by the axioms of $\mathrm{ZFC}$ together with large cardinal axioms. In this paper, we present other canonical extensions of $\mathrm{ZFC}$ that provide a strong structure theory for these classes. These axioms are variations of the Maximality Principle introduced by Stavi and Väänänen and later rediscovered by Hamkins.