ISL 1989 Problem 91

For $\phi: \mathbb{N} \mapsto \mathbb{Z}$ let us define $$M_{\phi} = \{f: \mathbb{N} \mapsto \mathbb{Z}, f(x) > f(\phi(x)), \forall x \in \mathbb{N} \}.$$ Prove that if $M_{\phi_1} = M_{\phi_2} \neq \emptyset,$ then $\phi_1 = \phi_2.$ Does this property remain true if $$M_{\phi} = \{f: \mathbb{N} \mapsto \mathbb{N}, f(x) > f(\phi(x)), \forall x \in \mathbb{N} \}?$$