Bipartite graphs with min weights

I have a full bipartite graph with node sets $ A=\{a_1,a_2,\ldots,a_n\}$ and $ B=\{b_1,b_2,\ldots,b_n\}$ . Each node has a weight, $ v_i$ for $ a_i$ and $ w_i$ for $ b_i$ . Each node $ a_i$ is connected to exactly one node of $ B$ , say $ b_j$ , via an edge $ e_i$ , whose weight is $ \min(v_i,w_j)$ . Now I want to find a one-to-one mapping from $ A$ to $ B$ , whose sum of edge weights is as small as possible.

My idea is to sort $ v_i$ s increasingly and $ w_i$ s decreasingly and then find the sum of all $ \min(v_i,w_i)$ after sorting. Is it correct? Can you give a proof/disproof?