Let $Z$ be a differentiable manifold and $Y$ a submanifold of $Z$, let $X \subset Y$. Prove let us $X$ is a submanifold of $Y$ if and only if $X$ is a submanifold of $Z$
suppose that Z is an n-dimensional manifold, Y is a k-dimensional submanifold and X is an l-dimensional submanifold. by definition for every $ x \in Y $ there is an open neighborhood $U$ of $Z$ and a chart $\varphi : U \to U' \subset \mathbb{R}^n$ such that $\varphi (U \cap Y)=\{u \in U': u_{k+1}=u_{k+2}=...=u_n=0\}$, i.e, $\varphi(U \cap Y)= \varphi (U) \cap \mathbb{R}^k$. Suposse that $X$ is a submanifold of $Y$, by definition for every $ x \in X $ there is an open neighborhood $V$ of $Y$ and a chart $\varphi_1 : V \to V' \subset \mathbb{R}^k$ such that $\varphi_1 (U \cap X)=\{u \in V': u_{l+1}=u_{l+2}=...=u_k=0\}$, i.e, $\varphi_1(V \cap X)= \varphi_1 (V) \cap \mathbb{R}^l$. To see that $ X $ is a submanifold of $Z$ we must find for each $x \in X$ a neighborhood $W$ of $x$ in $Z$ and a let chart $\rho:W \to W' \subset \mathbb{R}^n$ such that $\varphi (W \cap X)=\{u \in W': u_{l+1}=u_{l+2}=...=u_n=0\}$, i.e, $\varphi(W \cap Y)= \varphi (W) \cap \mathbb{R}^l$ , how can I proceed? any help would be appreciated.