Let us call a map $f: X \to Y$ between non-empty sets a "weak injection" if $f^{-1}(\{y\})\subseteq X$ is finite for every $y \in Y$. Consider the statement

Recall that the (Schroeder)-Cantor-Bernstein-Theorem (sometimes abbreviated by (CB)) states that if there are injections between two sets $X, Y$, then there is also a bijection between $X$ and $Y$. (CB) is a theorem of ${\sf (ZF)}$.

Consider the following statement:

(wiCB) If $X, Y$ are infinite sets such that there are weak injections between $X$ and $Y$, then there is a bijection between $X$ and $Y$.

Obviously, (wiCB) implies (CB). Can (wiCB) also be proved in ${\sf (ZF)}$?

Let us call a map $f: X \to Y$ between non-empty sets a "weak injection" if $f^{-1}(\{y\})\subseteq X$ is finite for every $y \in Y$.

Recall that the (Schroeder-)Cantor-Bernstein-Theorem (sometimes abbreviated by (CB)) states that if there are injections between two sets $X, Y$, then there is also a bijection between $X$ and $Y$. (CB) is a theorem of ${\sf (ZF)}$.

Consider the following statement:

(wiCB) If $X, Y$ are infinite sets such that there are weak injections between $X$ and $Y$, then there is a bijection between $X$ and $Y$.

Obviously, (wiCB) implies (CB). Can (wiCB) also be proved in ${\sf (ZF)}$?