Let $M\subset N$ be a submanifold. (Both $M$ and $N$ have no boundary, but $M\subset N$ need not to be closed as a subspace and none of them need to be compact) Choose a Riemannian metric on $N$ and identify the normal bundle $\vartheta M$ of $M$ with a subbundle of the tangent bundle of $N$ in the usual way. I'm searching for a reference with a detailed proof of the following fact:
There exists a open neighborhood of the zero section in $\vartheta_M$, such that the exponential map of $N$ provides a diffeomorphism of the neighborhood to an open neighborhood of $N$ in $M$.
A proof is given in Lang' "Fundamentals of Differential Geometry" in the case of $N\subset M$ closed as a subset. (Theorem 5.1 on page 110) A proof sketch of the general case is given as Theorem 6.5 of da Silva's Lectures on Symplectic Geometry.
I searched in almost every textbook about differential geometry and topology I know, but most authors prove the theorem for compact submanifolds or at most for topological closed ones. I even began to ask myself whether the statement may actually fail in the non closed case.