State and prove the $\epsilon$-Recursion Theorem. [You may assume the Principle of $\epsilon-$ Induction.]

What does it mean to say that a relation $r$ on a set $x$ is well-founded and extensional? State and prove Mostowski's Collapsing Theorem. [You may use any recursion theorem from the course, provided you state it precisely.]

For which sets $x$ is it the case that every well-founded extensional relation on $x$ is isomorphic to the relation $\epsilon$ on some transitive subset of $V_{\omega}$ ?