Theorem dflog2

Description: The natural logarithm function in terms of the exponential function restricted to its principal domain. (Contributed by Paul Chapman, 21-Apr-2008)

	Ref	Expression
Assertion	dflog2	|- log = `' ( exp |` ran log )

Proof

Step	Hyp	Ref	Expression
1		df-log	|- log = `' ( exp |` ( `' Im " ( -u _pi (,] _pi ) ) )
2		logrn	|- ran log = ( `' Im " ( -u _pi (,] _pi ) )
3	2	reseq2i	|- ( exp |` ran log ) = ( exp |` ( `' Im " ( -u _pi (,] _pi ) ) )
4	3	cnveqi	|- `' ( exp |` ran log ) = `' ( exp |` ( `' Im " ( -u _pi (,] _pi ) ) )
5	1 4	eqtr4i	|- log = `' ( exp |` ran log )