If $M$ and $N$ are Banach manifolds, $f:M\rightarrow N$ is a smooth map, and $q\in N$ is a regular value, so $f$ is a submersion on $f^{-1}(q)$, it is well known that the level set $f^{-1}(q)$ is a regular submanifold of $M$.

Question: Is there an analogous result for maps between manifolds modeled off locally convex spaces, in particular maps between smooth tame Frechet manifolds?

Question: Let $U$ be an open set in a smooth tame Frechet space $E$ (hence $U$ is a smooth tame Frechet manifold), $F$ be a smooth tame Frechet space, and $f:U\rightarrow F$ a smooth tame map. Is there a submersion or regular level set theorem in this context?