Theorem sscls

Description: A subset of a topology's underlying set is included in its closure. (Contributed by NM, 22-Feb-2007)

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

Proof

Step	Hyp	Ref	Expression
1		clscld.1	⊢ 𝑋 = ∪ 𝐽
2		ssintub	⊢ 𝑆 ⊆ ∩ { 𝑥 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑆 ⊆ 𝑥 }
3	1	clsval	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) = ∩ { 𝑥 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑆 ⊆ 𝑥 } )
4	2 3	sseqtrrid	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → 𝑆 ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) )