# Proving that the image of an injective, proper immersion is a manifold

I am trying to get through the proof of the statement "if $$f: M \to N$$ is injective, proper and an immersion, then $$f:M \to f(M)$$ is a diffeomorphism onto a submanifold".

The proof I'm reading says that on small sets $$U \subset M$$, $$f|_{U}: U \to f(U)$$ is a diffeomorphism onto its image, using the fact that for immersions $$f: M \to N$$ we can find coordinate charts around $$x$$ and $$f(x)$$ such that $$f(x_{1},\dots,x_{m})=(x_{1},\dots,x_{n},0,\dots,0)$$. However, I'm not sure how this fact about immersions would lead to $$f|_{U}$$ being a diffeomorphism for small sets U. I think I'm missing something basic but I don't know what it is.

Then, it says that if $$f(U)$$ is an open set of $$f(M)$$, then the inverse function is smooth, since it is continuous as $$f$$ is an open map (I understand this) and then that it is smooth by the inverse function theorem. I don't know how the inverse function theorem would give us this result (it says that for $$f: M \to N$$ smooth then if $$Df_{x}$$ is an isomorphism of tangent spaces then $$f$$ is locally a diffeomorphism). Thanks for the help.

The key fact here is that $$f$$ is proper and injective. In this case, one can check that $$f: M \to N$$ is an embedding as topological space. So now $$f(M)$$ is homeomorphic to $$M$$ and has a natural smooth structure: given a chart $$U$$ for $$M$$ , you declare $$f(U)$$ to be a chart for $$f(M)$$.
Now ,if you check the definition, this smooth structure implies that $$f:M \to f(M)$$ is smooth and in fact a diffeomorphism: it is a bijective open map being an homeomorphism as you correctly stated, so there is a continuous inverse $$g:f(M) \to M$$. The differential being invertible for every $$x \in M$$ (as you can check from the definition), there exists a local smooth inverse near $$f(x)$$, which must coincide with the global continous inverse $$g$$ , as $$f$$ is bijective.
Now, here it comes the immersion hypothesis. I don't know what is your definition of submanifold. The one I have in mind (which you can check it is equivalent to being the embedded(meaning immersion+topological embedding) image of a manifold) ,is that there are charts $$(U,\phi)$$ on $$N$$ such that $$\phi(U \cap f(M))=L \subset \mathbb{R}^n$$ ,with $$L$$ a vector subspace. As you can see, the immersion normal form, gives you this kind of chart for $$f(M)$$ and so you are done.