Write down the Axiom of Foundation.

What is the transitive closure of a set $x$ ? Prove carefully that every set $x$ has a transitive closure. State and prove the principle of $\in$-induction.

Let $(V, \in)$ be a model of $\mathrm{ZF}$. Let $F: V \rightarrow V$ be a surjective function class such that for all $x, y \in V$ we have $F(x) \in F(y)$ if and only if $x \in y$. Show, by $\in$-induction or otherwise, that $F$ is the identity.