[FOM] Restricted powersets for all sets

[Colin McLarty]

What is known about ZF without powerset but with an axiom "every set
has a set of all its countable subsets"?

This seems stronger than positing that the set of natural numbers has
a powerset, though I do not know a proof that it is.

More generally, for any definable cardinal \alpha, what about the
axiom "every set has a set of all smaller-than-\alpha subsets"?

best, Colin