In the professor Lee's introduction to smooth manifolds 2nd edition, the notion of diffeomorphism is defined for smooth manifolds with or without boundary. However, I saw some propositions that seems also to be true for smooth manifolds with boundary where professor Lee made the hypothesis only for smooth manifolds.
Example 2.14(b)on page 38: If $M$ is any smooth manifold and $(U,\varphi)$ is a smooth coordinate chart on $M$; then $\varphi:U\to \varphi(U)$ is a diffeomorphism.
Theorem 2.17 (Diffeomorphism Invariance of Dimension) on page 39: A nonempty smooth manifold of dimension $m$ cannot be diffeomorphic to an $n$-dimensional smooth manifold unless $m=n$.
Usually if some proposition is also true for the smooth manifolds with boundary, he will state clearly in the hypothesis. But this time he doesn't, so I wonder if there is any subtle reasons making the two propositions above no-longer true for smooth manifolds with boundary(I feel like a simple modification of the proof works also for the smooth manifolds with boundary), or professor Lee simply forgot to say "smooth manifolds with or without boundary"?