Implicit functions from locally convex spaces to Banach spaces

We first generalize the classical implicit function theorem of Hildebrandt and Graves to the case where we have a Keller ${\displaystyle C_{\Pi }^k}$-map f defined on an open subset of E×F and with values in F, for E an arbitrary Hausdorff locally convex space and F a Banach space. As an application, we prove that under a certain transversality condition the preimage of a submanifold is a submanifold for a map from a Fréchet manifold to a Banach manifold.