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Metamath Proof Explorer

Theorem fclsopni

Description: An open neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Mario Carneiro, 11-Apr-2015) (Revised by Stefan O'Rear, 8-Aug-2015)

Ref Expression
Assertion fclsopni ( ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) ∧ ( 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈 ∧ 𝑆 ∈ 𝐹 ) ) → ( 𝑈 ∩ 𝑆 ) ≠ ∅ )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ∪ 𝐽 = ∪ 𝐽
2 1 fclsfil ⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → 𝐹 ∈ ( Fil ‘ ∪ 𝐽 ) )
3 fclstopon ⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → ( 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) ↔ 𝐹 ∈ ( Fil ‘ ∪ 𝐽 ) ) )
4 2 3 mpbird ⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) )
5 fclsopn ⊢ ( ( 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) ∧ 𝐹 ∈ ( Fil ‘ ∪ 𝐽 ) ) → ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) ↔ ( 𝐴 ∈ ∪ 𝐽 ∧ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) )
6 4 2 5 syl2anc ⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) ↔ ( 𝐴 ∈ ∪ 𝐽 ∧ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) ) )
7 6 ibi ⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → ( 𝐴 ∈ ∪ 𝐽 ∧ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ) )
8 eleq2 ⊢ ( 𝑜 = 𝑈 → ( 𝐴 ∈ 𝑜 ↔ 𝐴 ∈ 𝑈 ) )
9 ineq1 ⊢ ( 𝑜 = 𝑈 → ( 𝑜 ∩ 𝑠 ) = ( 𝑈 ∩ 𝑠 ) )
10 9 neeq1d ⊢ ( 𝑜 = 𝑈 → ( ( 𝑜 ∩ 𝑠 ) ≠ ∅ ↔ ( 𝑈 ∩ 𝑠 ) ≠ ∅ ) )
11 10 ralbidv ⊢ ( 𝑜 = 𝑈 → ( ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ↔ ∀ 𝑠 ∈ 𝐹 ( 𝑈 ∩ 𝑠 ) ≠ ∅ ) )
12 8 11 imbi12d ⊢ ( 𝑜 = 𝑈 → ( ( 𝐴 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) ↔ ( 𝐴 ∈ 𝑈 → ∀ 𝑠 ∈ 𝐹 ( 𝑈 ∩ 𝑠 ) ≠ ∅ ) ) )
13 12 rspccv ⊢ ( ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ∀ 𝑠 ∈ 𝐹 ( 𝑜 ∩ 𝑠 ) ≠ ∅ ) → ( 𝑈 ∈ 𝐽 → ( 𝐴 ∈ 𝑈 → ∀ 𝑠 ∈ 𝐹 ( 𝑈 ∩ 𝑠 ) ≠ ∅ ) ) )
14 7 13 simpl2im ⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → ( 𝑈 ∈ 𝐽 → ( 𝐴 ∈ 𝑈 → ∀ 𝑠 ∈ 𝐹 ( 𝑈 ∩ 𝑠 ) ≠ ∅ ) ) )
15 ineq2 ⊢ ( 𝑠 = 𝑆 → ( 𝑈 ∩ 𝑠 ) = ( 𝑈 ∩ 𝑆 ) )
16 15 neeq1d ⊢ ( 𝑠 = 𝑆 → ( ( 𝑈 ∩ 𝑠 ) ≠ ∅ ↔ ( 𝑈 ∩ 𝑆 ) ≠ ∅ ) )
17 16 rspccv ⊢ ( ∀ 𝑠 ∈ 𝐹 ( 𝑈 ∩ 𝑠 ) ≠ ∅ → ( 𝑆 ∈ 𝐹 → ( 𝑈 ∩ 𝑆 ) ≠ ∅ ) )
18 14 17 syl8 ⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → ( 𝑈 ∈ 𝐽 → ( 𝐴 ∈ 𝑈 → ( 𝑆 ∈ 𝐹 → ( 𝑈 ∩ 𝑆 ) ≠ ∅ ) ) ) )
19 18 3imp2 ⊢ ( ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) ∧ ( 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈 ∧ 𝑆 ∈ 𝐹 ) ) → ( 𝑈 ∩ 𝑆 ) ≠ ∅ )