Theorem logf1o

Description: The natural logarithm function maps the nonzero complex numbers one-to-one onto its range. (Contributed by Paul Chapman, 21-Apr-2008)

	Ref	Expression
Assertion	logf1o	⊢ log : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log

Proof

Step	Hyp	Ref	Expression
1		eff1o2	⊢ ( exp ↾ ran log ) : ran log –1-1-onto→ ( ℂ ∖ { 0 } )
2		f1ocnv	⊢ ( ( exp ↾ ran log ) : ran log –1-1-onto→ ( ℂ ∖ { 0 } ) → ◡ ( exp ↾ ran log ) : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log )
3	1 2	ax-mp	⊢ ◡ ( exp ↾ ran log ) : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log
4		dflog2	⊢ log = ◡ ( exp ↾ ran log )
5		f1oeq1	⊢ ( log = ◡ ( exp ↾ ran log ) → ( log : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log ↔ ◡ ( exp ↾ ran log ) : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log ) )
6	4 5	ax-mp	⊢ ( log : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log ↔ ◡ ( exp ↾ ran log ) : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log )
7	3 6	mpbir	⊢ log : ( ℂ ∖ { 0 } ) –1-1-onto→ ran log