Theorem clsidm

Description: The closure operation is idempotent. (Contributed by NM, 2-Oct-2007)

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

Proof

Step	Hyp	Ref	Expression
1		clscld.1	⊢ 𝑋 = ∪ 𝐽
2	1	clscld	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) )
3	1	clsss3	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝑋 )
4	1	iscld3	⊢ ( ( 𝐽 ∈ Top ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝑋 ) → ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) ↔ ( ( cls ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) = ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) )
5	3 4	syldan	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) ↔ ( ( cls ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) = ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) )
6	2 5	mpbid	⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) = ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) )