Let $X$ denote a finite dimensional normed space.
A non empty set $M \subset X$ is called $d$-dimensional differentiable submanifold of $X$, if for all $a \in M$ there exists an open neighborhood $U$ of $a$ and a diffeomorphism $\varphi \colon U \to V$ with open $V \subset \mathbb{R}^n$, such that $$\varphi(M \cap U) = \mathbb{R}_0^d \cap V,$$ where $\mathbb{R}_0^d := \{x \in \mathbb{R^n} \mid x_{d+1} = \dots = x_n = 0\}$.
Why is $d$ well defined?
I know that if there exists a diffeomorphism between to normed spaces, they have to have the same dimension, since its differential induces an isomorphism. Hence, $\dim X = n$. But how does this help me in showing the uniqueness of $d$ (the book I am citing says it would)?