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Metamath Proof Explorer

Theorem fclstopon

Description: Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 26-Aug-2015)

Ref Expression
Assertion fclstopon ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ↔ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 fclstop ⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → 𝐽 ∈ Top )
2 istopon ⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ↔ ( 𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽 ) )
3 2 baib ⊢ ( 𝐽 ∈ Top → ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ↔ 𝑋 = ∪ 𝐽 ) )
4 1 3 syl ⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ↔ 𝑋 = ∪ 𝐽 ) )
5 eqid ⊢ ∪ 𝐽 = ∪ 𝐽
6 5 fclsfil ⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → 𝐹 ∈ ( Fil ‘ ∪ 𝐽 ) )
7 fveq2 ⊢ ( 𝑋 = ∪ 𝐽 → ( Fil ‘ 𝑋 ) = ( Fil ‘ ∪ 𝐽 ) )
8 7 eleq2d ⊢ ( 𝑋 = ∪ 𝐽 → ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ↔ 𝐹 ∈ ( Fil ‘ ∪ 𝐽 ) ) )
9 6 8 syl5ibrcom ⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → ( 𝑋 = ∪ 𝐽 → 𝐹 ∈ ( Fil ‘ 𝑋 ) ) )
10 filunibas ⊢ ( 𝐹 ∈ ( Fil ‘ ∪ 𝐽 ) → ∪ 𝐹 = ∪ 𝐽 )
11 6 10 syl ⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → ∪ 𝐹 = ∪ 𝐽 )
12 filunibas ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ∪ 𝐹 = 𝑋 )
13 12 eqeq1d ⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( ∪ 𝐹 = ∪ 𝐽 ↔ 𝑋 = ∪ 𝐽 ) )
14 11 13 syl5ibcom ⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 ) )
15 9 14 impbid ⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → ( 𝑋 = ∪ 𝐽 ↔ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) )
16 4 15 bitrd ⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ↔ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) )