This question is related to a question lately posted to $\cal MO$, as well as to this and that question, and especially the last where this post can be taken as a remedy of it since it was proved incconsistent.

So similarly, we add two partial unary functions $``j,f"$ to the language of $\sf ZF$.

The question is about if we can add the following on top of axioms of $\sf ZF$ [$j,f$ not used in Replacement nor Separation]?

$\exists \alpha: \text{ limit} (\alpha) \land j: V_{\alpha+1} \rightarrowtail V_\alpha \land f: V_\alpha \equiv rng(j) \\ \forall S: j[S]; j^{-1} [S] \text { both exist }\\\forall S: f[S]; f^{-1} [S] \text { both exist }\\ \forall S \in V_{\alpha +1}: j(S)=f[S] $

Where: "$\equiv$" signfy "bijection", and $g[S]=\{g(x) \mid x \in S\}$

The first three conditions have already been proved consistent, it's the addition of the fourth that is unsolved?