Metamath Proof Explorer


Theorem ntruni

Description: A union of interiors is a subset of the interior of the union. The reverse inclusion may not hold. (Contributed by Jeff Hankins, 31-Aug-2009)

Ref Expression
Hypothesis ntruni.1 𝑋 = 𝐽
Assertion ntruni ( ( 𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋 ) → 𝑜𝑂 ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ 𝑂 ) )

Proof

Step Hyp Ref Expression
1 ntruni.1 𝑋 = 𝐽
2 elssuni ( 𝑜𝑂𝑜 𝑂 )
3 sspwuni ( 𝑂 ⊆ 𝒫 𝑋 𝑂𝑋 )
4 1 ntrss ( ( 𝐽 ∈ Top ∧ 𝑂𝑋𝑜 𝑂 ) → ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ 𝑂 ) )
5 4 3expia ( ( 𝐽 ∈ Top ∧ 𝑂𝑋 ) → ( 𝑜 𝑂 → ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ 𝑂 ) ) )
6 3 5 sylan2b ( ( 𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋 ) → ( 𝑜 𝑂 → ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ 𝑂 ) ) )
7 2 6 syl5 ( ( 𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋 ) → ( 𝑜𝑂 → ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ 𝑂 ) ) )
8 7 ralrimiv ( ( 𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋 ) → ∀ 𝑜𝑂 ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ 𝑂 ) )
9 iunss ( 𝑜𝑂 ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ 𝑂 ) ↔ ∀ 𝑜𝑂 ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ 𝑂 ) )
10 8 9 sylibr ( ( 𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋 ) → 𝑜𝑂 ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ 𝑂 ) )