Suppose H = (W, F ).

Let f  be an isomorphism from G to H. Suppose that u ∈ V and v ∈ V. By the definition of an isomorphism,

 (1) {u,v} ∈ E ⇔ {f (u), f (v)} ∈ F.

Define

W₁ = f (V₁), the image of V₁ under f ,
W₂ = f (V₂), the image of V₂ under f .

Because f  is a bijection, (W₁, W₂) is a partition of the set W of vertices of H.

Choose any two vertices u and v in V₁. By the definitions of W₁, f (u) ∈ W₁ and f (v) ∈ W₁.

Since G is bipartite, {u,v} ∉ E. By (1), {f (u), f (v)} ∉ F. So there are no edges in H between two vertices in W₁.

A similar argument shows that there are no edges in H between two vertices in W₂.

So H is bipartite.