I was wondering what exactly "$A$ and $B$ are diffeomorphic" means (instead of "having a diffeomorphism $f:A\to B$). I know the definition of a diffeomorphism. For example, the set $A$ and $B$ are homeomorphic means that as topological spaces they are the same. I.e. topological property of $A$ and of $B$ are identical.
What would it be for diffeomorphism ? What for example it would mean that : if $f:\Omega \to \mathbb R^m$ is a $\mathcal C^1$ function where $\Omega \subset \mathbb R^n$ is open : there are two diffeomorphism $u: U\to P$ and $v: V\to Q$ (where $U\subset \Omega $, $P\subset \mathbb R^n$, $V,Q\subset \mathbb R^m $ open) s.t. $$v\circ f\circ u^{-1}(x_1,...,x_n)=(x_1,...,x_r,0,...,0).$$
I know that is (more or less) the constant rank theorem, but it's not the problem. Why the fact that $u$ and $v$ are diffeomorphism is important ? How can I interpret $$v\circ f\circ u^{-1}(x_1,...,x_n)=(x_1,...,x_r,0,...,0) \ \ ?$$ Would it be that $f$ is more or less the application $(x_1,...,x_n)\longmapsto (x_1,...,x_r,0,...,0)$ ? And if $u$ and $v$ where just bijection, why it would be bad ?