The ultraproduct is a well-defined nonempty structure

Equality sets show reflexivity because I is in U, symmetry directly, and transitivity because the intersection of the f=g and g=h sets is contained in the f=h set. The product is a set and is nonempty by F3; its equivalence classes and their quotient form sets by Separation and Replacement.

If each f_j is replaced by an equivalent g_j, intersect their finitely many equality sets to get E in U. On E, all function values and relation truth values agree. The function outputs are therefore equivalent by upward closure. For any two truth sets A,B agreeing on E, A in U implies $A\cap E\subseteq B$ and hence B in U; the converse is symmetric. Thus relations are independent as well. Empty arity gives E=I. Constant-symbol functions are uniquely specified. For the diagonal map, equality of [c_a] and [c_b] is equivalent to I in U when a=b and empty in U when a differs from b, proving injectivity.