Reachability relationship between a graph and its edge-reversed graph

Consider a directed graph $G = (V, E)$, where $V$ is the finite set of vertices and $E$ is the set of directed edges between the vertices. $G$ may contain cycles but there is no self-loop. Further, $G$ may not be strongly connected. Let $G^R$ be the graph obtained by reversing the directions of all the edges in $G$ without changing the set of vertices. Assume that Breadth First Search (BFS) or Depth First Search (DFS) from any given vertex $v$ of a graph visits only the reachable vertices from $v$ in that graph. Which of the following statements must always be true, regardless of the structure of $G$?