Theorem imf

Description: Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007) (Revised by Mario Carneiro, 6-Nov-2013)

	Ref	Expression
Assertion	imf	$$⊢\Im :ℂ⟶ℝ$$

Proof

Step	Hyp	Ref	Expression
1		df-im	$$⊢\Im =\left(x\in ℂ⟼\Re \left(\frac{x}{\mathrm{i}}\right)\right)$$
2		imval	$$⊢x\in ℂ\to \Im \left(x\right)=\Re \left(\frac{x}{\mathrm{i}}\right)$$
3		imcl	$$⊢x\in ℂ\to \Im \left(x\right)\in ℝ$$
4	2 3	eqeltrrd	$$⊢x\in ℂ\to \Re \left(\frac{x}{\mathrm{i}}\right)\in ℝ$$
5	1 4	fmpti	$$⊢\Im :ℂ⟶ℝ$$