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 ↾ ( ◡ ℑ “ ( - π (,] π ) ) )
2		logrn	⊢ ran log = ( ◡ ℑ “ ( - π (,] π ) )
3	2	reseq2i	⊢ ( exp ↾ ran log ) = ( exp ↾ ( ◡ ℑ “ ( - π (,] π ) ) )
4	3	cnveqi	⊢ ◡ ( exp ↾ ran log ) = ◡ ( exp ↾ ( ◡ ℑ “ ( - π (,] π ) ) )
5	1 4	eqtr4i	⊢ log = ◡ ( exp ↾ ran log )