FKMO 2013 Problem 6

For any permutation $f : \{ 1, 2, \cdots , n \} \to \{1, 2, \cdots , n \}$ , and define $$A = \{ i | i > f(i) \}$$ $$B = \{ (i, j) | i<j \le f(j) < f(i) \ or \ f(j) < f(i) < i < j \}$$ $$C = \{ (i, j) | i<j \le f(i) < f(j) \ or \ f(i) < f(j) < i < j \}$$ $$D = \{ (i, j) | i< j \ and \ f(i) > f(j)\}$$ Prove that $|A| + 2|B| + |C| = |D|$ .