Theorem clsss3

Description: The closure of a subset of a topological space is included in the space. (Contributed by NM, 26-Feb-2007)

	Ref	Expression
Hypothesis	clscld.1	⊢ 𝑋 = ∪ 𝐽
Assertion	clsss3	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		clscld.1	⊢ 𝑋 = ∪ 𝐽
2	1	clscld	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) )
3	1	cldss	⊢ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝑋 )
4	2 3	syl	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝑋 )