Metamath Proof Explorer


Theorem isopn3i

Description: An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016)

Ref Expression
Assertion isopn3i ( ( 𝐽 ∈ Top ∧ 𝑆𝐽 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) = 𝑆 )

Proof

Step Hyp Ref Expression
1 simpr ( ( 𝐽 ∈ Top ∧ 𝑆𝐽 ) → 𝑆𝐽 )
2 elssuni ( 𝑆𝐽𝑆 𝐽 )
3 eqid 𝐽 = 𝐽
4 3 isopn3 ( ( 𝐽 ∈ Top ∧ 𝑆 𝐽 ) → ( 𝑆𝐽 ↔ ( ( int ‘ 𝐽 ) ‘ 𝑆 ) = 𝑆 ) )
5 2 4 sylan2 ( ( 𝐽 ∈ Top ∧ 𝑆𝐽 ) → ( 𝑆𝐽 ↔ ( ( int ‘ 𝐽 ) ‘ 𝑆 ) = 𝑆 ) )
6 1 5 mpbid ( ( 𝐽 ∈ Top ∧ 𝑆𝐽 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) = 𝑆 )