# Need hint for bipartiteness proof

I am given a graph $$G = (V, E)$$ with $$N$$ connected components and $$G^\prime = (V^\prime, E^\prime)$$, where for each $$v \in V$$ there is $$v_1, v_2 \in V^\prime$$ and for each $$(u, v) \in E$$ there is $$(u_1, v_2), (u_2, v_1) \in E^\prime$$.

I need to prove:

$$G^\prime \textrm{ has 2N connected components} \Leftrightarrow G \textrm{ is bipartite}$$

I don’t know what technique or approach I need to use to prove this in either direction. I have a feeling I can use a direct proof for both directions. I have tried to come up with something but after an hour nothing has come up.