Let $\mathrm{R}_0,\cdots,\mathrm{R}_8$ be the following functions:

$\mathrm{R}_0(x,y)=\{x,y\}$

$\mathrm{R}_1(x,y)=x-y$

$\mathrm{R}_2(x)=\bigcup x$

$\mathrm{R}_3(x,y)=x\times y$

$\mathrm{R}_4(x)=\mathrm{Dom}(x)$

$\mathrm{R}_5(x)=\{(a,b)\ |\ a\in b\ \wedge\ a\in x\ \wedge\ b\in x\}$

$\mathrm{R}_6(x)=\{(a,b,c)\ |\ (b,a,c)\in x\}$

$\mathrm{R}_7(x)=\{(a,b,c)\ |\ (b,c,a)\in x\}$

$\mathrm{R}_8(x,y)=\{x[\{a\}]\ |\ a\in y\}\ \ $ where $x[\{a\}]=\{b\ |\ (a,b)\in x\}$ .

Gandy-Jensen system $\mathsf{GJ}_0$ is the system of which the set-theoretic axioms are the Axiom of Extensionality and the following nine set-existence axioms:

$\mathrm{R}_0(x,y)\in\mathrm{V},\ \cdots,\ \mathrm{R}_8(x,y)\in\mathrm{V}$ .

The class of rudimentary functions is the closure of $\mathrm{R}_0,\cdots,\mathrm{R}_8$ under composition.

It is obvious that if $F$ is a $n$-ary rudimentary function then $\mathsf{GJ}_0\vdash F(x_1,\cdots,x_n)\in\mathrm{V}$ .

My question is that whether the converse of the above proposition is true, i.e. is it true that:

For every class term $A(x_1,\cdots,x_n)$ whose free variables are in $x_1,\cdots,x_n$ , if $\mathsf{GJ}_0\vdash A(x_1,\cdots,x_n)\in\mathrm{V}$ , then $(x_1,\cdots,x_n)\mapsto A(x_1,\cdots,x_n)$ is a $n$-ary rudimentary function.

Edit 1 : I think I should make my terminologies more precise:

Let $\mathsf{T}$ be an axiom system of set theory (such as $\mathsf{GJ}_0,\ \mathsf{ZFC},$ and so on). We say that a function F is a n-ary $\mathsf{T}$-rudimentary function if there is a n-ary rudimentary function G such that $\mathsf{T}\vdash F=G$ .

I actually asked that whether the following statement is true:

For every class term $A(x_1,\cdots,x_n)$ whose free variables are in $x_1,\cdots,x_n$ , if $\mathsf{GJ}_0\vdash A(x_1,\cdots,x_n)\in\mathrm{V}$ , then $(x_1,\cdots,x_n)\mapsto A(x_1,\cdots,x_n)$ is a $n$-ary $\mathsf{ZFC}$-rudimentary function.

Where $\mathsf{ZFC}$ can be replaced by other systems such as $\mathsf{GJ}_0$ and then we have more similar questions.

Edit 2 : The question become trivial if I restrict "rudimentary" to "$\mathsf{T}$-rudimentary" defined in Edit 1 . So I will remove this restriction, and still ask that whether the original proposition (which is weaker than the false proposition in Edit 1 ) is true.