Arrays of distinct representatives — a very simple NP-complete problem

We prove that the following problem is NP-complete: Given an array (Aij), 1 ⩽ i ⩽ n, 1 ⩽ j ⩽ m of finite sets, does there exist an array of distinct representatives (xij) such that xij ϵ Aij, xij ≠ xik when j ≠ k, xij ≠ xkj when i ≠ k? The problem remains NP-complete even when n = 2, vbAijvb ⩽ 3, no element appears in more than four of the sets Aij, and there exist sets Bi, 1 ⩽ i ⩽ n and Ci, 1 ⩽ i ⩽ m such that Aij = Bi ∩ Cj for all i, j.