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Metamath Proof Explorer

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 ) )