Theorem dfii4

Description: Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015)

	Ref	Expression
Hypothesis	dfii4.1	$$⊢I={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\left[0,1\right]$$
Assertion	dfii4	$$⊢\mathrm{II}=\mathrm{TopOpen}\left(I\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfii4.1	$$⊢I={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\left[0,1\right]$$
2		eqid	$$⊢\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)=\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)$$
3	2	dfii3	$$⊢\mathrm{II}=\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right){\upharpoonright }_{𝑡}\left[0,1\right]$$
4	1 2	resstopn	$$⊢\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right){\upharpoonright }_{𝑡}\left[0,1\right]=\mathrm{TopOpen}\left(I\right)$$
5	3 4	eqtri	$$⊢\mathrm{II}=\mathrm{TopOpen}\left(I\right)$$