Absoluteness, idempotence and minimality of L

Internal ZF gives N its hierarchy history on each ordinal interval in N. External induction identifies its values: at zero both are empty; if the value at $\gamma$ is the actual $L_{\gamma }\in N$, F1 identifies its internal Def with $L_{\gamma +1}$. At a limit $\lambda \in N$, transitivity makes the internal history have every actual index $\gamma <\lambda$, and its internal union has exactly the union of their actual values. Thus $(L_{\alpha }{)}^N=L_{\alpha }\in N$ for every ordinal $\alpha$ of N.

If N contains all ordinals, every actual L level is therefore in N. Transitivity gives $L\subseteq N$, and the internal existential definition of constructibility ranges over precisely all actual ordinals, so its union is exactly L. For a set model the same argument stops at its ordinal height; no higher level is asserted to belong to N.

The six axioms in F2, Separation in F3, internal Power Set in F4, and Replacement in F5 establish all of ZF in L. Earlier level properties give transitivity and all ordinals. Thus L itself meets the hypotheses of step 2.1, which now yields $L^L=L$. Every element of L is internally constructible, exactly the relativization of $V=L$. This application occurs only after ZF in L has been established.