$$ \forall x \forall y ( x \ne y \implies f(x) \ne f(y)) \land \exists x \forall y (f(y) \ne x) $$
Is this first order equation satisfiable? Does there exist such a function? Because there can exist infinity of functions one can never prove that is unsatisfiable. My idea is, this is decidable but in infinity model. Is my way of thinking correct?