I am having trouble understanding the proof of
Theorem If $F : M \to N$ is an embedding, then $F (M)$ is an embedded submanifold of $N$
Proof. Let $p \in M$. Since $F$ is an immersion, by Theorem 1 (see below) there are charts $(U,\varphi)$ around $p$ and $(V,\psi )$ around $F(p)$ such that $F(U) \subseteq V$ and for all $x \in \varphi(U)$, $\psi\circ F \circ \varphi^{−1}(x) = (x; 0_{n−m})\tag{4}$
By applying translations we can assume that $\varphi(p) = 0_m$ and $\psi(F(p)) = 0_n$. Let $\varepsilon > 0$ so that
$B_{\varepsilon}(0_m) \subseteq \varphi(U)$ and $B_{\varepsilon}(0_n) \subseteq \psi(V )$
Then $B_{\varepsilon}(0_m) \times \{0_{n-m}\} \subseteq B_{\varepsilon}(0_n) \subseteq \psi(V)\tag{5}$
Because $F:M\to F(M) $is a homeomorphism, then $F(\varphi^{−1}(B_{\varepsilon}(0_m))) \subseteq F(M)$ open in the subspace topology. So there is an open subset $W \subseteq N$ such that
$F(\varphi^{−1}(B_{\varepsilon}(0_m))) =W\cap F(M) \tag{ 6}$
And because of (4),
$\psi(W\cap F(M)) = (\psi\circ F \circ \varphi^{−1})(B_{\varepsilon}(0_m))) \subseteq B_{\varepsilon}(0_m) \times \{0_{n-m}\}\tag{ 7}$
Let $\tilde {W}=W \cap \psi^{-1}(B_{\varepsilon}(0_n))$
So $(\tilde {W}, \psi|_{\tilde W}) $ is a chart around $F(p)$ with
$\psi|_{\tilde {W}}(\tilde{W} \cap F(M)) $
$= \psi|_{\tilde{W}}( W \cap \psi^{-1}(B_{\varepsilon}(0_n)) \cap F(M))$ $=\psi (W \cap F(M)) \tag{8}$ $=B_{\varepsilon}(0_m)\times \{0_{n-m}\}\tag{9}$ $=\psi|_{\tilde {W}}(\tilde {W})\cap (\Bbb R^{m}\times\{0_{n-m}\})\tag{10}$
1)Is there is a typo in (7)? I think it should be "$=$" instead of "$\subseteq$" because then I think that's what it's being used to go from (8) to (9).
2)I can't figure out what is going on in the last chain of equalities.The preceding expressions should be used somehow , but it's not clear how. How does $\psi^{-1}(B_{\varepsilon}(0_n))$ dissapear in (8)? How do I get (9)? and how do I go from there to (10)?
The definition of embedded submanifolds that is being used in this last part is:
DEF Let $M$ be a smooth manifold of dimension $m$, and $S ⊂ M$. Then $S$ is an embedded submanifold of dimension $k$ if for every $p \in S $ there is a chart $(U, \varphi)$ of $M$ around $p$ is such that $\varphi(U \cap S) = (\Bbb R^k \times \{0_{m-k}\}) \cap \varphi(U)$
and at the beginning of the proof a particular case of the rank theorem used:
Theorem 1 (Rank theorem for injective differential). Suppose $M$ is a smooth manifold of dimension $m$, and that $N$ is a smooth manifold of dimension $n$. Suppose $F : M \to N$ is smooth. Let $p \in M$. If $dF_p$ is injective, then there are charts $(U, \varphi)$ of M around p and $(V,\psi )$ of N around $F(p)$ such that $F(U) \subseteq V$ and for all $x \in\varphi(U)$, and
$\psi\circ F \circ \varphi^{−1}(x) = (x, 0_{n−m})$