I am trying to prove the following statement:
Let $f:X\to Y$ be a embedding of differentiable manifolds. Then there is a unique manifold structure on $S:=f(X)$ such that the inclusion map $S \hookrightarrow Y$ is a diffeomorphism.
In Lee's book "Introduction to Smooth Manifolds", he argues in Theorem 5.21 as follows: Suppose $\tilde{S}$ were some other smooth manifold structure on $S$ such that the inclusion $\tilde{i}: \tilde{S} \hookrightarrow Y$ is an embedding. Then the restricted map $\tilde{i}:\tilde{S} \to S$ is a smooth map. Since $\tilde{i}: S\to Y$ has injective derivative maps, so does $\tilde{i}:\tilde{S} \to S$. But then $\tilde{i}:\tilde{S} \to S$ is a bijective immersion, so is a diffeomorphism. The part in bold follows from what he calls the "Global Rank Theorem" (his Theorem 4.14). This theorem uses the Baire Category Theorem and relies on $\tilde{S}$ being second-countable.
Is there any simplification of this argument? Using the BCT feels a little like taking a sledgehammer to a walnut. If one knows about homology, then we can argue as follows: $\tilde{i}: \tilde{S} \to Y$ is a homeomorphism onto its image $S$ so (since $\mathbb{R}^n \not \cong \mathbb{R}^m$ for $n\neq m$ by homology) $\tilde{S},S$ have the same dimension, whence $d_x\tilde{i}$ is bijective for all $x\in \tilde{S}$. Then $\tilde{i}$ is a bijective local diffeomorphism so is a diffeomorphism.
However, now we are reliant on the machinery of homology instead. Is there any truly elementary argument, which is only "smooth manifolds"-y in nature?