Let $M$ be a countable transitive model of $\sf ZF$, Let $V\_\alpha, V\_{\alpha+1}$ be two stages of the cumulative hierarchy in $M$, let $f: V\_\alpha \to V\_{\alpha+1}$ be a bijection such that for any set $S \in V\_{\alpha+2}$ we have both $f[S] \in M ; f^{-1}[S] \in M$ . Let $F: V\_{\alpha+1} \to V\_{\alpha+2}; F(x)=f[x]=\{f(y) \mid y \in x\}$.



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> Is there a proof that we may have a set $S \in V\_{\alpha+3}$ such that we don't have both $F[S] \in M; F^{-1}[S] \in M$?
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Is it possible to have such nesting of functions?
I think I have an answer to this question of mine. if $M$ in the question satisfy $AC$, then there must exist some set $S$ in $V\_{\alpha+3}$ that doesn't have both $F[S]; F^{-1}[S]$ being elements of $M$, otherwise we can interpret $\sf NF+AC$ which is inconsistent. However, the proof of that result is long.


