Suppose that $g$ has a domain set $E$ and a codomain set $F$. If $E$ is the empty set, then $g$ is injective due to
R2752: Empty map is injection, so we may assume that $E$ is nonempty. Let $x, y \in E$ such that $g(x) = g(y)$. Since $g$ is a submap of $f$, then $f(x) = g(x) = g(y) = f(y)$. Since $f$ is an injection, it follows that $x = y$. Since $x, y \in E$ were arbitrary, $g$ is an injection. $\square$
