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 ) ≠ ∅ )