At some point in my work (which has nothing to do with set theoretics foundation) I need to consider the following axiom: 



> 
> For any set $X$, any class $V$ with a surjective map $f : V \twoheadrightarrow X$ there exists a small subclass $V' \subset V$
>  such that the restriction of $f$ to $V'$ is already surjective.
> 
> 
> 


(The general framework I'm working in is intuitionist mathematics with a notion of class, and small for a class mean in bijection with a set.)


It can be considered as a weak form of the axiom of choice, as the class valued axiom of choice would be about finding a $V'$ such that the restriction of $f$ is bijective (and in particular $V'$ would be small)


What I want to know is if this axiom appear somewhere in the literature. and especially does it have a name ? As it been studied or used ?


Edit : Eric Wofsey pointed out in the comment that in ZF (with the regularity axioms) this 'axiom' is in fact a theorem. This unfortunately does not apply to the framework I had in mind, but highly suggest that this probably does not have a name or hasn't been studied as an axiom anywhere...



Does this axiom (a weak form of class valued choice) has a name?
In weak set theories, using classical logic and interpreting
"small subclass" as "set", this principle amounts to an
alternative formulation of the collection axiom. For example, in
Zermelo set theory or even much weaker theories, even without the
power set axiom, this principle is equivalent to the collection
axiom scheme.


**Collection:** If $X$ is a set and $\forall a\in X\exists b\
\varphi(a,b)$, then there is a set $Y$ such that $\forall a\in
X\exists b\in Y\ \varphi(a,b)$.


One can replace the talk of a formula $\varphi(a,b)$ with
membership in a class of pairs.


If we have collection, then suppose we have a class surjective
function $f:V\to X$ as in your case. Since for every $a\in
X\exists b\ f(b)=a$, by collection we can find a set $Y$ such that
for every $a\in X\exists b\in Y\ f(b)=a$, and so $f\upharpoonright
Y:Y\to X$ is already surjective, as desired.


Conversely, if your axiom holds and we have a formula
$\varphi(a,b)$ for an instance of collection, so that $\forall
a\in X\exists b\ \varphi(a,b)$, then let $f(a,b)=a$, provided that
$\varphi(a,b)$, so that $f$ is surjective from the class
$\{(a,b)\mid \varphi(a,b)\}$ to $X$. Under your axiom, there is a
set $Y$ such that $f\upharpoonright Y:Y\to X$ is already
surjective. In this case, the projection of $Y$ onto the second
coordinate, that is, the set $B=\{b\mid \exists a\ (a,b)\in Y\}$
is a set, and the surjectivity of the restriction amounts to
$\forall a\in X\exists b\in B\ \varphi(a,b)$, thus verifying the
desired instance of collection.


This argument doesn't seem to need much at all in the background
set theory, although I am not sure what the effects would be
without classical logic.


