I'm working through Lee's Differential Geometry textbook and have run into the regular level set theorem:
Theorem: Every regular level set of a smooth map between smooth manifolds is a properly embedded submanifold whose codimension is equal to the dimension of the codomain.
What I understand this to mean is that given two smooth manifold $U, V$, and a smooth mapping between them (call it $\phi:U\rightarrow V$), then for all $c \in V$ such that $d\phi${$\phi^{-1}(c)$} is surjective (note {$\phi^{-1}(c)$} is most likely a set of points, not a singleton), that that level set {$\phi^{-1}(c)$} is a properly embedded submanifold.
I'm just wondering how to use this. For example, One problem I was looking at was to show that a set $\tilde{G}(n,k)$ described to be the set of $n\times n$ matrices $M$ that are idempotent, symmetric, and have trace$(M)=k$, is a smooth manifold using the level set theorem. I know that this set is bijective to the Grassmannian of $\mathbb{R}^n$ of dimension $k$ (linear subspaces of dimension $k$ of $\mathbb{R}^n$) but I have no clue how to proceed from here.
When and how is the level set theorem useful in situations like these?