I'm really blocked in the first part of the below exercise and can't solve it. It would be great if you could help!
Show that there exists a partial recursive function $g:\mathbb{N}\to \mathbb{N}$ such that : $$W_i \neq \emptyset \implies g(i) \in W_i$$ Moreover, show that there exists a total recursive function $\gamma$ such that : $$ W_{\gamma(i)}\subset W_i \text{ and } (W_i \neq \emptyset \implies card(W_{\gamma(i)}) = 1) $$ Then, show that given a partial recursive function $f:\mathbb{N}\to \mathbb{N}$ then there exists a total function $a$ such that : $$ f(x) \downarrow \implies W_{a(x)} = \bigcup_{i \leq f(x)} W_{\gamma(i)}$$
N.B. $f(x) \downarrow$ translates to $f$ is defined on $x$, and $W_i$ is the definitional domain of the universal function $\phi_i$.