  The mapping class group $MCG(\Sigma_g)$ of a surface of genus $g$ has a
long-history in topology and group theory. More recently, the mapping class
group $MCG(V_g)$ of a handlebody $V_g$ of genus $g$ has become an interesting
topic in the study of $3$ manifolds, largely thanks to Heegaard splitting.
While $MCG(V_g)$ can be regarded naturally as a sub group of $MCG(\Sigma_g)$,
we could not find any complete proof of this fundamental theorem. It is the
purpose of this paper that we give a rigorous proof of embedding of $MCG(V_g)$
into $MCG(V_g)$. The key step is: Any self-homeomorphism $f$ of handlebody
$V_g$ of genus $g$ is ambient isotopic to identity if the restriction
$f|_{\partial V_g}$ is isotopic to identity.
