MathDB

For

\phi: \mathbb{N} \mapsto \mathbb{Z}

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

Problem Statement