Is the implicit function theorem needed to relate the EMP to UMP? My professor says the expenditure function and the indirect utility function are inverses of each other. But how is this possible? Consider each.

$$v:(p_x,p_y,m) \rightarrow \Bbb{R}$$ 

$$e:(p_x,p_y,\bar{U}) \rightarrow \Bbb{R}$$ 

These are both mappings from $\Bbb{R^3}\rightarrow \Bbb{R}$. How can one be the inverse of the other? 

Only if one was a scalar field and the other a vector field would this work.

My Question:

What is the relationship between the expenditure minimization function and the indirect utility function?
Fix prices at some arbitrary positive values $p_x^0,p_y^0$ and view the functions $v$ and $e$ as $v:\mathbb{R}\rightarrow \mathbb{R}$ and $e:\mathbb{R}\rightarrow \mathbb{R}$. Then they are inverses.