Metamath Proof Explorer


Theorem eff1o2

Description: The exponential function restricted to its principal domain maps one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 21-Apr-2008) (Revised by Mario Carneiro, 13-May-2014)

Ref Expression
Assertion eff1o2
|- ( exp |` ran log ) : ran log -1-1-onto-> ( CC \ { 0 } )

Proof

Step Hyp Ref Expression
1 logrn
 |-  ran log = ( `' Im " ( -u _pi (,] _pi ) )
2 1 eff1o
 |-  ( exp |` ran log ) : ran log -1-1-onto-> ( CC \ { 0 } )