Club agreement of countable filtrations ($f[F_{\delta }]=G_{\delta }\text{on a club}$)

For two kappa-filtrations at $\kappa ={\omega }_1$ exhausting sets of size ${\aleph }_1$, any bijection between the sets carries corresponding stages onto one another on a club set of indices. Bound both forward and inverse images at each countable stage by a function $g:{\omega }_1\to {\omega }_1$. The limit ordinals closed under all earlier $g$-values form a club; continuity gives equality at those ordinals.