Absoluteness of names and their ranks

Induct externally in actual membership rank on $\tau \in M$. Whether every entry is a pair with second coordinate in P is absolute by transitivity and F3. Every first coordinate sigma then belongs to M and has smaller membership rank. By induction sigma is internally a name iff it is actually a name. F1 characterizes namehood by exactly these entry conditions in both universes; its stage-bound converse is available internally in M because M satisfies ZF. Thus namehood agrees, including the empty name.

For a name tau, its internal set of predecessor ranks plus one exists by internal Replacement. The induction hypothesis identifies each predecessor rank, and F2 and F3 identify ordinal successors and the union giving the supremum. Both rank recursions consequently give the same value at tau. The empty predecessor set gives zero on both sides. No comparison of the full power sets at a name stage entered either induction.