Theorem dfii5

Description: The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015)

	Ref	Expression
Assertion	dfii5	⊢ II = ( ordTop ‘ ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfii2	⊢ II = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] 1 ) )
2		unitssre	⊢ ( 0 [,] 1 ) ⊆ ℝ
3		eqid	⊢ ( ordTop ‘ ≤ ) = ( ordTop ‘ ≤ )
4		eqid	⊢ ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) )
5	3 4	xrrest	⊢ ( ( 0 [,] 1 ) ⊆ ℝ → ( ( ordTop ‘ ≤ ) ↾t ( 0 [,] 1 ) ) = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] 1 ) ) )
6	2 5	ax-mp	⊢ ( ( ordTop ‘ ≤ ) ↾t ( 0 [,] 1 ) ) = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] 1 ) )
7		ordtresticc	⊢ ( ( ordTop ‘ ≤ ) ↾t ( 0 [,] 1 ) ) = ( ordTop ‘ ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) )
8	1 6 7	3eqtr2i	⊢ II = ( ordTop ‘ ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) )