Theorem flimsncls 23374

Description: If 𝐴 is a limit point of the filter 𝐹, then all the points which specialize 𝐴 (in the specialization preorder) are also limit points. Thus, the set of limit points is a union of closed sets (although this is only nontrivial for non-T1 spaces). (Contributed by Mario Carneiro, 20-Sep-2015.)

Assertion

Ref	Expression
flimsncls	⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ (𝐽 fLim 𝐹))

Proof of Theorem flimsncls

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

Step	Hyp	Ref	Expression
1		flimtop 23353	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2		eqid 2731	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	flimelbas 23356	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 ∈ ∪ 𝐽)
4	3	snssd 4774	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → {𝐴} ⊆ ∪ 𝐽)
5	2	clsss3 22447	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽) → ((cls‘𝐽)‘{𝐴}) ⊆ ∪ 𝐽)
6	1, 4, 5	syl2anc 584	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ ∪ 𝐽)
7	6	sselda 3947	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 ∈ ∪ 𝐽)
8		simpll 765	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝐴 ∈ (𝐽 fLim 𝐹))
9	8, 1	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝐽 ∈ Top)
10		simprl 769	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝑦 ∈ 𝐽)
11	1	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐽 ∈ Top)
12	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → {𝐴} ⊆ ∪ 𝐽)
13		simpr 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 ∈ ((cls‘𝐽)‘{𝐴}))
14	11, 12, 13	3jca 1128	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → (𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})))
15	2	clsndisj 22463	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → (𝑦 ∩ {𝐴}) ≠ ∅)
16		disjsn 4677	. . . . . . . . . . 11 ⊢ ((𝑦 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝑦)
17	16	necon2abii 2990	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑦 ↔ (𝑦 ∩ {𝐴}) ≠ ∅)
18	15, 17	sylibr 233	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝐴 ∈ 𝑦)
19	14, 18	sylan 580	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝐴 ∈ 𝑦)
20		opnneip 22507	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
21	9, 10, 19, 20	syl3anc 1371	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
22		flimnei 23355	. . . . . . 7 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 ∈ 𝐹)
23	8, 21, 22	syl2anc 584	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝑦 ∈ 𝐹)
24	23	expr 457	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ 𝑦 ∈ 𝐽) → (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))
25	24	ralrimiva 3139	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → ∀𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))
26		toptopon2 22304	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
27	11, 26	sylib 217	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐽 ∈ (TopOn‘∪ 𝐽))
28	2	flimfil 23357	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
29	28	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐹 ∈ (Fil‘∪ 𝐽))
30		flimopn 23363	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐹 ∈ (Fil‘∪ 𝐽)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))))
31	27, 29, 30	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))))
32	7, 25, 31	mpbir2and 711	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 ∈ (𝐽 fLim 𝐹))
33	32	ex 413	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑥 ∈ ((cls‘𝐽)‘{𝐴}) → 𝑥 ∈ (𝐽 fLim 𝐹)))
34	33	ssrdv 3953	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ (𝐽 fLim 𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060   ∩ cin 3912   ⊆ wss 3913  ∅c0 4287  {csn 4591  ∪ cuni 4870  ‘cfv 6501  (class class class)co 7362  Topctop 22279  TopOnctopon 22296  clsccl 22406  neicnei 22485  Filcfil 23233   fLim cflim 23322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-fbas 20830  df-top 22280  df-topon 22297  df-cld 22407  df-ntr 22408  df-cls 22409  df-nei 22486  df-fil 23234  df-flim 23327

This theorem is referenced by:  tsmscls  23526