That task is, that given a k-dimensional $C^1$-submanifold of $\mathbb{R}^n$ and $\phi \colon \mathbb{R}^n \rightarrow\mathbb{R}^n$ a diffeomorphism. Show, that $M' := \phi(M)$ also is a $k$-dimensional submanifold of $\mathbb{R}^n$.
In our lecture we have already shown, that: $M$ is a $k$-dimensional manifold in $C^\mathcal{l} \ \Leftrightarrow$ For every $x_0 \in M$, there is a neighbourhood $U \subset \mathbb{R}^n$, another neighbourhood $V \subset \mathbb{R}^k$ of $0$ and a regular $C^l$-function $\phi \colon V \rightarrow U$ such that $M \cap U = \phi(V), \ \phi(0) = x_0$.
My idea is to use that theorem and show that those things exist for $M'$. I think it would work if we use the $\phi'$ that we get from $M$, since that is a manifold. Now $\phi \circ \phi'$ would satisfy the conditions of the theorem when we also define $V' := V$ and $U' := \phi'(U)$. To show the other things would simple then.
Is this the right idea? I feel like it should be pretty simple.