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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	$⊢ ({\exp ↾}_{ran log}) : ran log \underset{1-1 onto}{⟶} ℂ ∖ \{0\}$

Proof

Step	Hyp	Ref	Expression
1		logrn	$⊢ ran log = ℑ^{-1} [(- π, π]]$
2	1	eff1o	$⊢ ({\exp ↾}_{ran log}) : ran log \underset{1-1 onto}{⟶} ℂ ∖ \{0\}$