I am trying to solve Problem 5-14 from Lee's Smooth Manifolds. Please check if the reasoning is tight.
Problem: Suppose $M$ is a smooth manifold and $S\subset M$ is an immersed submanifold. For the given topology on $S$, there is only one smooth structure making $S$ into an immersed submanifold.
the proof:
Take $S'$ to be the same set as $S$ with the same topology and add some smooth structure on it. Denote $i,i'$ to be the inclusions of $S,S'$ into $M$ and $f$ the inclusion of $S'$ into $S$. This inclusion $f$ is continuous because $S,S'$ have the same topologies. Restriction to submanifolds then implies that $f$ is a smooth map. $i\circ f = i'$ holds. The composition $di'_p$: $T_pS' \to T_pS \to T_pM$ implies then that $df_p$ is injective. $f$ is then an immersion and also bijective, hence a diffeomorphism. This means that $S$ and $S'$ have the same smooth structure.