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Metamath Proof Explorer

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	$⊢ log : ℂ ∖ \{0\} \underset{1-1 onto}{⟶} ran log$

Proof

Step	Hyp	Ref	Expression
1		eff1o2	$⊢ ({\exp ↾}_{ran log}) : ran log \underset{1-1 onto}{⟶} ℂ ∖ \{0\}$
2		f1ocnv	$⊢ ({\exp ↾}_{ran log}) : ran log \underset{1-1 onto}{⟶} ℂ ∖ \{0\} \to {({\exp ↾}_{ran log})}^{-1} : ℂ ∖ \{0\} \underset{1-1 onto}{⟶} ran log$
3	1 2	ax-mp	$⊢ {({\exp ↾}_{ran log})}^{-1} : ℂ ∖ \{0\} \underset{1-1 onto}{⟶} ran log$
4		dflog2	$⊢ log = {({\exp ↾}_{ran log})}^{-1}$
5		f1oeq1	$⊢ log = {({\exp ↾}_{ran log})}^{-1} \to (log : ℂ ∖ \{0\} \underset{1-1 onto}{⟶} ran log \leftrightarrow {({\exp ↾}_{ran log})}^{-1} : ℂ ∖ \{0\} \underset{1-1 onto}{⟶} ran log)$
6	4 5	ax-mp	$⊢ log : ℂ ∖ \{0\} \underset{1-1 onto}{⟶} ran log \leftrightarrow {({\exp ↾}_{ran log})}^{-1} : ℂ ∖ \{0\} \underset{1-1 onto}{⟶} ran log$
7	3 6	mpbir	$⊢ log : ℂ ∖ \{0\} \underset{1-1 onto}{⟶} ran log$