This question is regarding p.223-224 of Loring Tu's Introduction to Manifolds (Second edition).
Without proof, the author previously assumed the following.
Theorem. Let $M$ be a manifold and $X$ be a smooth vector field on it. For any $p \in M$ and a chart $(U, x^{1}, \cdots, x^{n})$ there is a neighborhood $W \ni p$ in $U$ and a smooth map (called a local flow) $F : (-\epsilon, \epsilon) \times W \rightarrow U$ such that for each $q \in W$, the mapping $F_{q} : W \rightarrow U$ given by $t \mapsto F(t, q)$ defines an integral curve of $X$ starting at $q$ (i.e., $F(0, q) = q$ and $d/dt|_{t = t_{0}}F(t, q) = X_{F(t_{0}, q)}$ for any $t_{0} \in (-\epsilon, \epsilon)$).
In p.223, the author says that for $p \in M$, we may choose a neighborhood $U \ni p$ in $M$ with a local flow $\varphi : (-\epsilon, \epsilon) \times U \rightarrow M$ such that $(\partial/\partial t) \varphi_{t}(q) = X_{\varphi_{t}(q)}$ and $\varphi_{0}(q)$ for all $q \in U$, where $\varphi_{t}(q) := \varphi(t, q)$.
In one of the earlier sections, the book explained that $\varphi_{s} \circ \varphi_{t} = \varphi_{s + t}$, whenever both sides make sense by uniqueness of integral curves of a vector field. So far, I am comfortable understanding what the book is trying to say.
Here is what I don't understand.
Where I need help. In p.224, the author says that the formula $\varphi_{s} \circ \varphi_{t} = \varphi_{s + t}$ implies that $\varphi_{t} : U \rightarrow \varphi_{t}(U)$ is a diffeomorphism because $\varphi_{-t} \circ \varphi_{t} = \varphi_{0} = id$ and $\varphi_{t} \circ \varphi_{-t} = \varphi_{0} = id$. I am concerned with the domain of $\varphi_{-t}$ because it does not look trivial to me that it is defined on $\varphi_{t}(U)$. It is also strange that in the line "$\varphi_{-t} \circ \varphi_{t} = \varphi_{0} = id$ and $\varphi_{t} \circ \varphi_{-t} = \varphi_{0} = id$," the identity must be the same ones, which seems to imply that $\varphi_{t}(U) = U$, and again, I do not see any reason that this is true.
I suppose I have this difficulty since I do not understand bolts and nuts of the existence of local flow, and at this moment, it seems to take a long roundabout from studying manifolds to read the proof of it. Thus, I ask you for a generous help in explaining why $\varphi_{t} : U \rightarrow \varphi_{t}(U)$ is a diffeomorphism with inverse $\varphi_{-t}$. Thank you in advance.
The proof may be pretty long and boring (it belongs to ODE books), but the result you need is easy to state:
Let's see how the above helps. Let $p\in \varphi_t(U)$ be any point. Pick $q\in U$ such that $\varphi_t(q)=p$. The time-shifted map $ s\mapsto \varphi_{t+s}(q)$, $-t\le s\le 0$ is also a trajectory of $X$ (since $X$ is time-independent). We can denote this map $\varphi_s(p)$, since $\varphi_0(p)=p$. So, $(p,-t)\in E$.
By the openness of $E$, $\varphi_{-t}$ is defined in a neighborhood of $p$. By the continuity, it maps a small neighborhood of $p$ into $U$. Hence, this neighborhood lies in $\varphi_t(U)$, showing that $\varphi_t(U)$ is open.
To summarize: we've seen that $\varphi_t$ maps $U$ onto an open set $\varphi_t(U)$, then $\varphi_{-t}$ is defined on $\varphi_t(U)$. That the compositions $\varphi_t \circ \varphi_{-t} $ and $\varphi_{-t} \circ \varphi_{t} $ bring us back to the point we started with is a direct consequence of the definition: we follow the same curve back and forth.