Suppose that $\phi:H^{n} \cap V \rightarrow M$ is a diffeomorphism between a relatively open subspace of the half-space $H^{n}$ and a set $M$, so that the set $M$ is an $n$-manifold with boundary (I'm doing the "with boundary" case for generality). Suppose that $C \subseteq M$ is such that the subset $\phi^{-1}(C) \subseteq H^{n} \cap V$ is an $m-$ manifold with boundary. That is, there exists a diffeomorphism $\phi':H^{m} \cap V' \rightarrow \phi^{-1}(C)$.
Because the compositions of diffeomorphisms is a diffeomorphism, it should then follow that
$$\phi \circ \phi' : H^{m}\cap V' \rightarrow C$$
is a diffeomorphism, so that $C$ is also an $m-$ manifold (with boundary). Is this correct?
Putting this all together, we'd have that if $\phi:H^{n} \cap V \rightarrow M$ is an $n-$manifold with boundary and $S \subseteq H^{n} \cap V$ is an $m-$dimensional subspace (in the manifold sense), then $\phi(S)$ is an $m-$dimensional subspace as well. So, "diffeomorphisms preserve the dimension of subsets".
I thank you all in advance!
The claim that "an −manifold (with boundary), by definition, it is diffeomorphic to $H^m$" is quite false. What is true is that every point of an $m$-dimensional manifold with boundary has a neighborhood which is diffeomorphic to an open subset of $H^m$.
Other than that, yes, $C$ is an $m$-dimensional submanifold with boundary, simply because diffeomorphisms (such as $\phi^{-1}$) send submanifolds (with boundary) to submanifolds with boundary, and dimension is preserved. (Saying that something is an $m$-dimensional subspace in a manifold is weaker. The notion of dimension, say, the covering dimension, is defined for arbitrary topological spaces, not just for manifolds. For manifolds with boundary, it equals the dimension in the sense of the definition of a manifold.) Incidentally, I suggest you avoid saying that a subset of a manifold is a manifold, since it is ambiguous; instead, use the terminology "submanifold."