"Subpaths" of Dijkstra's shortest path also shortest?
Of course, it is true. However, it is not a property of Dijkstra's algorithm but is property of the shortest paths themselves. Suppose that is not true, and you have a shorter path from $C$ to $B$, we denote it as $p_{CB}$. Then we would have a path $A \rightarrow p_{CB} \rightarrow F \rightarrow E$ from $A$ to $E$ which is shorter than $A  \rightarrow  C  \rightarrow  D  \rightarrow B  \rightarrow  F  \rightarrow E$.