Let $P \subset \mathbb{N}$ be a finite index set and suppose that there is a partial function $f:P \to P$, which is undefined for some strict subset of $P$.
Then, suppose I have a function $g: P \to \{ 0, 1 \}$ defined as $$ g(i) = \begin{cases} 0 & f(i) \text{ is undefined} \\ 1 & f(i) \text{ is defined.} \end{cases} $$
What is the formally correct way to denote these predicates? Could $\not \exists f(i)$ and $\exists f(i)$ work?