Let $M$ be a smooth $n$-manifold with boundary. Suppose $U\subset \mathbb{R}^{n-1}$ is open, $a,b$ are real numbers with $a<b$, and $F:(a,b]\times U \to M$ is a smooth embedding that restricts to an embedding of $\{b\}\times U$ into $\partial M$.
In this case, how can we show that $F$ is a local diffeomorphism? This question arises from Lemma 15.27 of Lee's Introduction to Smooth Manifolds, where he considers $M$ to be oriented and shows that $F$ is orientation-preserving for $M$ if and only if $f:U \to \partial M$ given by $f(x)=F(b,x)$ is orientation-preserving for $\partial M$. The definition of an orientation-preserving map from the book stipulates that the map is a local diffeomorphism.
Since we know that smooth embeddings are diffeomorphisms onto its image as a subspace topology, it suffices to show that $F((a,b]\times U)=:S$ is an open subset of $M$. But I am not sure how to prove this. Since $U$ and $\partial M$ are manifolds without boundary, we know that $f$ being an embedding is a local diffeomorphism. So my idea is that $F$ should map points $(a,b)\times U$ to the manifold interior of $M$, and so $f(U)=F(\{b\}\times U)$ being open in $\partial M$, should give a coordinate chart $(x^1,\dots , x^{n-1})$ around every $f(x)=F(b,x)$ and we can extend this chart to $(x^1,\dots, x^{n-1}, x^n)$, where $x^n \in (a,b]$, on the half space $\mathbb{H}^n$, but I can't figure out how to formalize this and guarantee that $F$ maps $(a,b)\times U$ into the interior. I would greatly appreciate any help.
