For context, I am reading Appendix F is Milnor's book on Sullivan's No Wandering Theorem and this is where he uses the following manifolds result:
Let $F:X \rightarrow Y$ be a smooth function between manifolds of dimension $n$ and $m$ respectively where $n > m$. If there exists some $x \in X$ such that the rank of $DF\vert_x$ takes a maximal value $r \leq m$, then there exists a neighbourhood $N$ of $x$ such that $F(N) \subset Y$ is a submanifold of dimension $r$.
I don't believe I've seen this result before but it looks somewhat like the inverse function theorem. If anyone could illuminate where this result is from, it would be greatly appreciated.
Thanks
This is in fact a direct application of the implicit function theorem, which is usually mentioned in a single breath with the inverse function theorem. The implicit function theorem gives conditions under which a level set of a smooth function is locally the graph of a smooth function. Now, in your case it is useful to note that graphs of smooth functions are in fact (embedded) submanifolds.
Some authors (such as Forster or Kühnel, both of whom publish mainly in German however) actually choose to define submanifolds directly as follows
A $k$-dimensional submanifold $M \subset \mathbb{R}^n$ is defined as being the level set $F^{-1}(0)$ of a smooth map $$ \mathbb{R}^n \supset U \stackrel{F}{\longrightarrow} \mathbb{R}^{n-k} $$ with maximal rank. I.e. $\text{rank}(DF_x) = n-k$ for every $x \in M \cap U$, where $M \cap U = F^{-1}(0)$ for a suitable neighborhood $U$ for every point in $M$. Locally one can then describe $M$ as the image of an immersion $$ \mathbb{R}^k \supset V \stackrel{f}{\longrightarrow} M \subset \mathbb{R}^n$$ with $\text{rank}(Df) = k$.
If you want to read up more on this, I highly suggest Lee's "Introduction to smooth manifolds," which is an excellent textbook on differential geometry. If by any chance you do speak German, I can also recommend Kühnel's "Differentialgeometrie," where the approach you mentioned is used to introduce submanifolds, although they only play a small role in the rest of the text.