= Relative condensation lemma
If $\alpha$ is a nonzero <limit ordinal> and $Y\prec L_\alpha(X)$ with $X\cup\{X\}\subseteq Y$, the <Mostowski collapse theorem> sends $Y$ to $L_\beta(X)$ for some nonzero limit $\beta\leq\alpha$ and fixes $X$ pointwise. The base is fixed by transitivity and inclusion of all its members. Transfer the <relative constructible level recognition> formula first by elementarity and then through the collapse isomorphism to identify its image. If $\rho=X\cap\operatorname{Ord}$, ordinal heights are $\rho+\alpha$ and $\rho+\beta$, giving the bound. Having only $X\in Y$ does not guarantee this base-fixing version.
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