Theorem ioontr

Description: The interior of an interval in the standard topology on RR is the open interval itself. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	ioontr	⊢ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 (,) 𝐵 ) ) = ( 𝐴 (,) 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		iooretop	⊢ ( 𝐴 (,) 𝐵 ) ∈ ( topGen ‘ ran (,) )
2		retop	⊢ ( topGen ‘ ran (,) ) ∈ Top
3		ioossre	⊢ ( 𝐴 (,) 𝐵 ) ⊆ ℝ
4		uniretop	⊢ ℝ = ∪ ( topGen ‘ ran (,) )
5	4	isopn3	⊢ ( ( ( topGen ‘ ran (,) ) ∈ Top ∧ ( 𝐴 (,) 𝐵 ) ⊆ ℝ ) → ( ( 𝐴 (,) 𝐵 ) ∈ ( topGen ‘ ran (,) ) ↔ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 (,) 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) ) )
6	2 3 5	mp2an	⊢ ( ( 𝐴 (,) 𝐵 ) ∈ ( topGen ‘ ran (,) ) ↔ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 (,) 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) )
7	1 6	mpbi	⊢ ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 (,) 𝐵 ) ) = ( 𝐴 (,) 𝐵 )