Use of inverse in defining Lebesgue measure

Question

Reading literature on the development of the Lebesgue integral, I encountered statements similar to this: Let $$E_i=\{x\in [a,b]: y_i\leq f(x) \leq y_{i+1}\} = f^{-1}([y_j, y_{j+1})) $$ I understand the first part (set of all points x corresponding to a slice of the function range). What confuses me is the use of the inverse. Does the second part imply the function f is required to be invertible on the given interval? Or is this a different notion of "inverse"?