Theorem fclsopn 23939

Description: Write the cluster point condition in terms of open sets. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
fclsopn	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))

Distinct variable groups:   𝑜,𝑠,𝐴   𝑜,𝐹,𝑠   𝑜,𝐽,𝑠   𝑜,𝑋,𝑠

Proof of Theorem fclsopn

Step	Hyp	Ref	Expression
1		isfcls2 23938	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
2		filn0 23787	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
3	2	adantl 481	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ≠ ∅)
4		r19.2z 4446	. . . . . 6 ⊢ ((𝐹 ≠ ∅ ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))
5	4	ex 412	. . . . 5 ⊢ (𝐹 ≠ ∅ → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
6	3, 5	syl 17	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
7		topontop 22838	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
8	7	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → 𝐽 ∈ Top)
9		filelss 23777	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ 𝑋)
10	9	adantll 714	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ 𝑋)
11		toponuni 22839	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
12	11	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → 𝑋 = ∪ 𝐽)
13	10, 12	sseqtrd 3968	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ ∪ 𝐽)
14		eqid 2733	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
15	14	clsss3 22984	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑠) ⊆ ∪ 𝐽)
16	8, 13, 15	syl2anc 584	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → ((cls‘𝐽)‘𝑠) ⊆ ∪ 𝐽)
17	16, 12	sseqtrrd 3969	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → ((cls‘𝐽)‘𝑠) ⊆ 𝑋)
18	17	sseld 3930	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ 𝑋))
19	18	rexlimdva 3135	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ 𝑋))
20	6, 19	syld 47	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ 𝑋))
21	20	pm4.71rd 562	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
22	7	ad3antrrr 730	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → 𝐽 ∈ Top)
23	13	adantlr 715	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ ∪ 𝐽)
24		simplr 768	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → 𝐴 ∈ 𝑋)
25	11	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → 𝑋 = ∪ 𝐽)
26	24, 25	eleqtrd 2835	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → 𝐴 ∈ ∪ 𝐽)
27	14	elcls 22998	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ ∪ 𝐽 ∧ 𝐴 ∈ ∪ 𝐽) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅)))
28	22, 23, 26, 27	syl3anc 1373	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅)))
29	28	ralbidva 3155	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑠 ∈ 𝐹 ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅)))
30		ralcom 3262	. . . . 5 ⊢ (∀𝑠 ∈ 𝐹 ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅) ↔ ∀𝑜 ∈ 𝐽 ∀𝑠 ∈ 𝐹 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅))
31		r19.21v 3159	. . . . . 6 ⊢ (∀𝑠 ∈ 𝐹 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅) ↔ (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))
32	31	ralbii 3080	. . . . 5 ⊢ (∀𝑜 ∈ 𝐽 ∀𝑠 ∈ 𝐹 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))
33	30, 32	bitri 275	. . . 4 ⊢ (∀𝑠 ∈ 𝐹 ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))
34	29, 33	bitrdi 287	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))
35	34	pm5.32da 579	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴 ∈ 𝑋 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
36	1, 21, 35	3bitrd 305	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058   ∩ cin 3898   ⊆ wss 3899  ∅c0 4284  ∪ cuni 4860  ‘cfv 6489  (class class class)co 7355  Topctop 22818  TopOnctopon 22835  clsccl 22943  Filcfil 23770   fClus cfcls 23861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-iin 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-fbas 21298  df-top 22819  df-topon 22836  df-cld 22944  df-ntr 22945  df-cls 22946  df-fil 23771  df-fcls 23866

This theorem is referenced by:  fclsopni  23940  fclselbas  23941  fclsnei  23944  fclsbas  23946  fclsss1  23947  fclsrest  23949  fclscf  23950  isfcf  23959