Theorem ntrss3

Description: The interior of a subset of a topological space is included in the space. (Contributed by NM, 1-Oct-2007)

	Ref	Expression
Hypothesis	clscld.1	⊢ 𝑋 = ∪ 𝐽
Assertion	ntrss3	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		clscld.1	⊢ 𝑋 = ∪ 𝐽
2	1	ntropn	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ∈ 𝐽 )
3	1	eltopss	⊢ ( ( 𝐽 ∈ Top ∧ ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ∈ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝑋 )
4	2 3	syldan	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝑋 )