Theorem clsss2

Description: If a subset is included in a closed set, so is the subset's closure. (Contributed by NM, 22-Feb-2007)

	Ref	Expression
Hypothesis	clscld.1	⊢ 𝑋 = ∪ 𝐽
Assertion	clsss2	⊢ ( ( 𝐶 ∈ ( Clsd ‘ 𝐽 ) ∧ 𝑆 ⊆ 𝐶 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		clscld.1	⊢ 𝑋 = ∪ 𝐽
2		cldrcl	⊢ ( 𝐶 ∈ ( Clsd ‘ 𝐽 ) → 𝐽 ∈ Top )
3	2	adantr	⊢ ( ( 𝐶 ∈ ( Clsd ‘ 𝐽 ) ∧ 𝑆 ⊆ 𝐶 ) → 𝐽 ∈ Top )
4	1	cldss	⊢ ( 𝐶 ∈ ( Clsd ‘ 𝐽 ) → 𝐶 ⊆ 𝑋 )
5	4	adantr	⊢ ( ( 𝐶 ∈ ( Clsd ‘ 𝐽 ) ∧ 𝑆 ⊆ 𝐶 ) → 𝐶 ⊆ 𝑋 )
6		simpr	⊢ ( ( 𝐶 ∈ ( Clsd ‘ 𝐽 ) ∧ 𝑆 ⊆ 𝐶 ) → 𝑆 ⊆ 𝐶 )
7	1	clsss	⊢ ( ( 𝐽 ∈ Top ∧ 𝐶 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝐶 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝐶 ) )
8	3 5 6 7	syl3anc	⊢ ( ( 𝐶 ∈ ( Clsd ‘ 𝐽 ) ∧ 𝑆 ⊆ 𝐶 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝐶 ) )
9		cldcls	⊢ ( 𝐶 ∈ ( Clsd ‘ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ 𝐶 ) = 𝐶 )
10	9	adantr	⊢ ( ( 𝐶 ∈ ( Clsd ‘ 𝐽 ) ∧ 𝑆 ⊆ 𝐶 ) → ( ( cls ‘ 𝐽 ) ‘ 𝐶 ) = 𝐶 )
11	8 10	sseqtrd	⊢ ( ( 𝐶 ∈ ( Clsd ‘ 𝐽 ) ∧ 𝑆 ⊆ 𝐶 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝐶 )