Prove the existence of a one-to-one function for a recursive definition

Question

Specifically, I don’t know what $g(x) = \{f(x)\}$ means, thus I’m not sure how to proceed with the proof. Is it implied that I need to define another function $f(x)$ or do I need to prove the latter $f(x)$?

Answer 1

Now you want to define a function $g:A\to\mathcal{P}(Y)$, that is, for any element $x\in A$, you need to associate a subset of $Y$. The subset is chosen to be the singleton $\{f(x)\}$.

It is of course a well-defined function. The problem is to show that $g$ is one-to-one. However, since $f$ is one-to-one, we have $g(x)=g(x')\implies \{f(x)\}=\{f(x')\}\implies f(x)=f(x')\implies x=x'$.