Hints only.
My claim is yes.
Let's say there's two graphs, A and B.
Assuming A is bipartite, A can then be split up into two different graphs a1 and a2. There may be edges between vertices in a1 and a2, but not between members of the same group (no a1 vertice is connected to another vertice in a1).
Isomorphism is a one-to-one and onto function by nature, so every element in A is uniquely related to an element in B. That being said, we can split up graph B into two parts as well - b1 (the set of vertices related to by elements in a1) and b2 ( the set of vertices related to by elements in b2).
Since relationships between vertices are preserved, every element in b1 should therefore be connected only to elements in b2, and not with each other thus making it a bipartite graph.
Am I on the right track?