Atiyah-J\"anich theorem for $\sigma$-C*-algebras

K-theory for $\sigma$-C*-algebras (countable inverse limits of C*-algebras) has been investigated by N. C. Phillips [{\it K-Theory} {\bf 3} (1989), 441–478]. We use his representable K-theory to show that the space of Fredholm modular operators with coefficients in an arbitrary unital $\sigma$-C*-algebra $A$, represents the functor $X\mapsto {\mathrm{R}\mathrm{K}}_0(C(X,A))$ from the category of countably compactly generated spaces to the category of abelian groups.