Theorem flimsncls 23961

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 23940	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2		eqid 2737	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	flimelbas 23943	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 ∈ ∪ 𝐽)
4	3	snssd 4753	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → {𝐴} ⊆ ∪ 𝐽)
5	2	clsss3 23034	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽) → ((cls‘𝐽)‘{𝐴}) ⊆ ∪ 𝐽)
6	1, 4, 5	syl2anc 585	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ ∪ 𝐽)
7	6	sselda 3922	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 ∈ ∪ 𝐽)
8		simpll 767	. . . . . . 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 771	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝑦 ∈ 𝐽)
11	1	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐽 ∈ Top)
12	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → {𝐴} ⊆ ∪ 𝐽)
13		simpr 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 ∈ ((cls‘𝐽)‘{𝐴}))
14	11, 12, 13	3jca 1129	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → (𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})))
15	2	clsndisj 23050	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → (𝑦 ∩ {𝐴}) ≠ ∅)
16		disjsn 4656	. . . . . . . . . . 11 ⊢ ((𝑦 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝑦)
17	16	necon2abii 2983	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑦 ↔ (𝑦 ∩ {𝐴}) ≠ ∅)
18	15, 17	sylibr 234	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝐴 ∈ 𝑦)
19	14, 18	sylan 581	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝐴 ∈ 𝑦)
20		opnneip 23094	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
21	9, 10, 19, 20	syl3anc 1374	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
22		flimnei 23942	. . . . . . 7 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 ∈ 𝐹)
23	8, 21, 22	syl2anc 585	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝑦 ∈ 𝐹)
24	23	expr 456	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ 𝑦 ∈ 𝐽) → (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))
25	24	ralrimiva 3130	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → ∀𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))
26		toptopon2 22893	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
27	11, 26	sylib 218	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐽 ∈ (TopOn‘∪ 𝐽))
28	2	flimfil 23944	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
29	28	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐹 ∈ (Fil‘∪ 𝐽))
30		flimopn 23950	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐹 ∈ (Fil‘∪ 𝐽)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))))
31	27, 29, 30	syl2anc 585	. . . 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 714	. . 3 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 ∈ (𝐽 fLim 𝐹))
33	32	ex 412	. 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑥 ∈ ((cls‘𝐽)‘{𝐴}) → 𝑥 ∈ (𝐽 fLim 𝐹)))
34	33	ssrdv 3928	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ (𝐽 fLim 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  {csn 4568  ∪ cuni 4851  ‘cfv 6492  (class class class)co 7360  Topctop 22868  TopOnctopon 22885  clsccl 22993  neicnei 23072  Filcfil 23820   fLim cflim 23909

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-fbas 21341  df-top 22869  df-topon 22886  df-cld 22994  df-ntr 22995  df-cls 22996  df-nei 23073  df-fil 23821  df-flim 23914

This theorem is referenced by:  tsmscls  24113