It is well known that if a smooth Map $f : M \to N$ between two smooth manifolds (finite dimensional) is transversal to a submanifold $L \subset N, L \pitchfork f$, than $f^{-1}(N)$ is a submanifold of $M$.
According to the german wikipedia entry about transversality, the converse (if the preimage under a smooth map $f$ of a submanifold is also a submanifold, the map is transversal to N, so $f \pitchfork N$) is also true (and I didn’t find a proof of this).
I ask myself, whether that is true? According to Serge Lang the usual definition involving the complimentary tangent spaces is equivalent to: A certain composition of maps is a submersion. Thus I would have to prove that this composition is a submersion, i.e. has regular values.
EDIT: So apparently, in general this is false. To avoid confusion, I removed a part of the question describing a situation I thought I was finding myself in, which was not the case. I leave the general part of the question for anyone asking themselves this question.