Theorem ref

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

	Ref	Expression
Assertion	ref	$$⊢\Re :ℂ⟶ℝ$$

Proof

Step	Hyp	Ref	Expression
1		df-re	$$⊢\Re =\left(x\in ℂ⟼\frac{x+\bar{x}}{2}\right)$$
2		reval	$$⊢x\in ℂ\to \Re \left(x\right)=\frac{x+\bar{x}}{2}$$
3		recl	$$⊢x\in ℂ\to \Re \left(x\right)\in ℝ$$
4	2 3	eqeltrrd	$$⊢x\in ℂ\to \frac{x+\bar{x}}{2}\in ℝ$$
5	1 4	fmpti	$$⊢\Re :ℂ⟶ℝ$$