Theorem iscld3

Description: A subset is closed iff it equals its own closure. (Contributed by NM, 2-Oct-2006)

	Ref	Expression
Hypothesis	clscld.1	⊢ 𝑋 = ∪ 𝐽
Assertion	iscld3	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑆 ∈ ( Clsd ‘ 𝐽 ) ↔ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) = 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		clscld.1	⊢ 𝑋 = ∪ 𝐽
2		cldcls	⊢ ( 𝑆 ∈ ( Clsd ‘ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) = 𝑆 )
3	1	clscld	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) )
4		eleq1	⊢ ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) = 𝑆 → ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) ↔ 𝑆 ∈ ( Clsd ‘ 𝐽 ) ) )
5	3 4	syl5ibcom	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) = 𝑆 → 𝑆 ∈ ( Clsd ‘ 𝐽 ) ) )
6	2 5	impbid2	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑆 ∈ ( Clsd ‘ 𝐽 ) ↔ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) = 𝑆 ) )