 Relative to a model \(M\), this \(\mathbf{ESO}\) sentence asserts the existence of functions \(f\) and \(g\) and an element \(t\) such that \(f = g\) (implication from left to right), this function is injective (implication from right to left), and its domain is the whole domain of \(M\) but the element \(t\) does not appear in its range. Consequently the range is a proper subset of the domain of \(M\). In other words, the sentence \(\phi_{inf}\) is true in a model \(M\) iff the domain of \(M\) is infinite.