Boundedly finite-to-one functions

A function is boundedly finite-to-one if there is a natural number k such that each point has at most k inverse images. In this paper, we prove in ZF (i.e., the Zermelo--Fraenkel set theory without the axiom of choice) several results concerning this notion, among which are the following: (1) For each infinite set A and natural number n, there is no boundedly finite-to-one function from S(A) to S≤ n(A), where S(A) is the set of all permutations of A and S≤ n(A) is the set of all permutations of A moving at most n points. (2) For each infinite set A, there is no boundedly finite-to-one function from B(A) to fin(A), where B(A) is the set of all partitions of A such that every block is finite and fin(A) is the set of all finite subsets of A.