The exercise is:
Let $\varphi\newcommand{\dom}{\operatorname{dom}}$ be a formula. There is a Gödel operation $G$ such that for every transitive $U$ and all $x_{1}, \ldots, x_{n} \in U$ $$ \{u \in U : U \models \varphi (u,x_{1}, \ldots, x_{n})\}= G(U,x_{1}, \ldots, x_{n}) $$ [Hint: use the normal form theorem]
I tryed to solve this exercise but I don't know how to write a formal solution.
We call $X_{i}= \{x_{i}\}\forall i=1,...,n$. So by the normal form theorem there is a Gödel operation $G$ such that $G(U,X_{1}, \dotsc , X_{n})=\{(u,x_{1},\dotsc,x_{n}) : u \in U\wedge\varphi(u, x_{1}, \dotsc , x_{n}) \}$. The set that we are looking for is $$\underbrace{\dom \dotso\dom}_{\text{$n$ times}} (G(U,X_{1}, \dotsc , X_{n})).$$ We used both $\{x,y\}$ and $\dom(X)$ that are Gödel operations.
My problem is that the set that I built should be $\{u \in U : \varphi (u,x_{1}, \ldots, x_{n})\}$ which is a little bit different from $\{u \in U : U \models \varphi (u,x_{1}, \dotsc, x_{n})\}$. That is because the formula from which we start is not a restricted formula so, theoretically, we can't use the normal form theorem.
So how can I fix it?
Instead of using the original formula, use its relativization to $U$. In addition to giving you what you want at the end, since $U$ is a set, the relativization is $\Delta_0,$ which is required for the normal form theorem.