The critical point of a measurable ultrapower

U is countably complete since kappa is uncountable, so F1 applies. For any nonempty set x with size eta<kappa, choose a bijection b from eta to x using F3. A predecessor [f] E [c_x] has f(i) in x on a U-large set. Replace f outside that set by b(0), without changing its class. Composing with the inverse of b gives a function to eta, hence has a constant U-large fibre by F2. Therefore [f]=[c_y] for some y in x. Conversely every y in x gives such a predecessor. For empty x there are no predecessors by properness. The collapse equation now gives $j(x)=\{j(y):y\in x\}$ whenever |x|<kappa.

Every ordinal alpha<kappa has size below kappa. By induction, step 1.1 gives $j(\alpha )=\{j(\beta ):\beta <\alpha \}=\alpha$, including alpha=0. The identity function always takes values in kappa, so its collapsed class d belongs to j(kappa), hence is an ordinal. For every alpha<kappa the tail $\{\xi :\alpha <\xi <\kappa \}$ is U-large, since its complement has size below kappa by F2. Thus alpha=j(alpha) belongs to d. Consequently $\kappa \le d<j(\kappa )$; kappa is moved and all earlier ordinals are fixed.

By F2 kappa is inaccessible and by F4 every x in V_kappa has size below kappa. Apply rank induction to such x. Every y in x has smaller rank and remains in V_kappa, so the induction hypothesis fixes y. Step 1.1 then gives $j(x)=\{j(y):y\in x\}=x$. The induction starts with empty and includes all limit ranks without a separate choice of representatives.