Theorem flimclsi 24164

Description: The convergent points of a filter are a subset of the closure of any of the filter sets. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Assertion

Ref	Expression
flimclsi	⊢ (𝑆 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆))

Proof of Theorem flimclsi

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	flimfil 24155	. . . . . . 7 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
3	2	ad2antlr 740	. . . . . 6 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝐹 ∈ (Fil‘∪ 𝐽))
4		flimnei 24153	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦 ∈ 𝐹)
5	4	adantll 727	. . . . . 6 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦 ∈ 𝐹)
6		simpll 779	. . . . . 6 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆 ∈ 𝐹)
7		filinn0 24046	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐽) ∧ 𝑦 ∈ 𝐹 ∧ 𝑆 ∈ 𝐹) → (𝑦 ∩ 𝑆) ≠ ∅)
8	3, 5, 6, 7	syl3anc 1398	. . . . 5 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → (𝑦 ∩ 𝑆) ≠ ∅)
9	8	ralrimiva 3159	. . . 4 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦 ∩ 𝑆) ≠ ∅)
10		flimtop 24151	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
11	10	adantl 487	. . . . 5 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝐽 ∈ Top)
12		filelss 24038	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐽) ∧ 𝑆 ∈ 𝐹) → 𝑆 ⊆ ∪ 𝐽)
13	12	ancoms 464	. . . . . 6 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝐹 ∈ (Fil‘∪ 𝐽)) → 𝑆 ⊆ ∪ 𝐽)
14	2, 13	sylan2 605	. . . . 5 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑆 ⊆ ∪ 𝐽)
15	1	flimelbas 24154	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 ∈ ∪ 𝐽)
16	15	adantl 487	. . . . 5 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 ∈ ∪ 𝐽)
17	1	neindisj2 23309	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ∪ 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦 ∩ 𝑆) ≠ ∅))
18	11, 14, 16, 17	syl3anc 1398	. . . 4 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦 ∩ 𝑆) ≠ ∅))
19	9, 18	mpbird 260	. . 3 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘𝑆))
20	19	ex 418	. 2 ⊢ (𝑆 ∈ 𝐹 → (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
21	20	ssrdv 3944	1 ⊢ (𝑆 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  {csn 4591  ∪ cuni 4874  ‘cfv 6540  (class class class)co 7416  Topctop 23079  clsccl 23204  neicnei 23283  Filcfil 24031   fLim cflim 24120

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-fbas 21548  df-top 23080  df-cld 23205  df-ntr 23206  df-cls 23207  df-nei 23284  df-fil 24032  df-flim 24125

This theorem is used by:  flimcls  24171  flimfcls  24212  cnextcn  24253  cmetss  25504  minveclem4  25620