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Metamath Proof Explorer

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 ) ) ) )