 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 $\beta \le \alpha$ 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 $\rho =X\cap \mathrm{Ord}$, ordinal heights are $\rho +\alpha$ and $\rho +\beta$, giving the bound. Having only $X\in Y$ does not guarantee this base-fixing version.