I know that my question is closely related to this topic: Restriction of a smooth vector bundle is a smooth bundle?
That is, given a vector bundle show that for a immersed/embedded submanifold the restriction defines a vector bundle by using the vector bundle chart lemma.
However, if I use the vector bundle chart lemma (as noted in the text) I really don't see any step wherein I need the injectivity of the differential map. That is, isn't a smooth mapping on M: $$\tau_{\alpha, \beta}(p) \rightarrow GL(k,\mathbb{R})$$ still smooth when I introduce a "submanifold" S such that the inclusion mapping $S \mapsto M$ is smooth? In that sense, I don't understand where the condition on the rank of the differential is needed. It would be great if someone could point out my misunderstanding in the proof!