# 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?