Define wholly stratified $\sf NF$ to be $\sf NF$ with its language restricted to stratified expressions.
In this theory we can arrive at a general implementation of tuples, that is:
$\langle x_1,..,x_n\rangle = ((\iota^{-t_1}(x_1),.., \iota^{-t_n}(x_n)) , (-t_1,..,-t_n))$
where: $\iota^0(x)=x \\ \iota^{n+1}(x)=\{\iota^n(x)\}$
$ R= \{ \langle x_1,..,x_n\rangle\mid \phi(x_1,..,x_n) \}$
Here $t_i$ is the type of the variable $x_i$ in formula $\phi(x_1,..,x_n)$ according to stratification standards but with fixing the highest type to be $0$, then obtain lower types by $-1$ steps. $(,..,)$ stands for Quine-Rosser tuples, which are extensions of Quine-Rosser pairs to cover tuples with more than two projections. This is defined as:
$(x_1,..,x_n) = \overset n {\underset {i=1} \bigcup} \psi_n (x_i)$
$\psi_n(x_i) = \varphi_n ``x_i \cup \{ j \in \mathbb N \mid j < i-1 \}$
$\varphi_n(x) = y \iff (x \in \mathbb N \land y=x+n-1) \land (x \not \in \mathbb N \land y=x) $
$\varphi`` x = \{\varphi(y) \mid y \in x\}$
This way ANY formula $\phi$ of the language of this theory with free variables, would define some $n$-ary relation (or function) that can be implemented by a set.
So all relations and functions definable in that langauge are representable as Sets!
Is there a comparable situation with the standard line of set theories, I mean $\sf ZC, ZFC, NBG, MK, ..$, I mean a fragment or an extension of them that enable one to implement ALL functions and relations in its language in terms of sets.
