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 ?

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...