In this posting what was termed as "Proper Class Choice" principle turned to be equivalent to Global Choice over the base theory of "MK-Foundation -Limitation of size + Set Replacement*". However if the restriction to relations with set domains was placed (as in this posting), then it became equivalent to "set choice + collection" over the above base theory. This made me think of the following restriction:
Axiom of Small Class Choice :$$\forall \ relation \ R^{ |dom| < |V|} \exists F \subset R \ [F: dom(R) \to rng(R)]$$
In English: for every relation from a small class domain, there is a subclass of it that is a function from the same domain.
A class is said to be small if and only if it is strictly smaller in cardinality than the universe $V$ of all sets.
$^*$ Set Replacement is the assertion: "replacement classes from sets, are sets".
Questions
Are there models of the "base theory + small class choice" in which there exist small proper classes?
If yes, is small class choice principle stronger than "choice over sets + collection" over the base theory? I mean are there models satisfying the base theory in which choice over sets + collection is satisfied but small class choice fails?
