“Subpaths” of Dijkstra's shortest path also shortest?

I have trouble putting this into formal mathematical terms, so let's suppose that I found the shortest path from A to E as A > C > D > B > F > E with Dijkstra's algorithm. Am I correct in assuming that the shortest path from C to B would be C > D > B, and that the shortest path from D to F would be D > B > F?

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 $$A \rightarrow C \rightarrow D \rightarrow B \rightarrow F \rightarrow E$$.