Theorem flimsncls 23873

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 23852	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2		eqid 2729	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	flimelbas 23855	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 ∈ ∪ 𝐽)
4	3	snssd 4773	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → {𝐴} ⊆ ∪ 𝐽)
5	2	clsss3 22946	. . . . . 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 3946	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 ∈ ∪ 𝐽)
8		simpll 766	. . . . . . 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 770	. . . . . . . 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 1128	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → (𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})))
15	2	clsndisj 22962	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → (𝑦 ∩ {𝐴}) ≠ ∅)
16		disjsn 4675	. . . . . . . . . . 11 ⊢ ((𝑦 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝑦)
17	16	necon2abii 2975	. . . . . . . . . 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 580	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝐴 ∈ 𝑦)
20		opnneip 23006	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
21	9, 10, 19, 20	syl3anc 1373	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
22		flimnei 23854	. . . . . . 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 456	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ 𝑦 ∈ 𝐽) → (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))
25	24	ralrimiva 3125	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → ∀𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))
26		toptopon2 22805	. . . . . 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 23856	. . . . . 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 23862	. . . . 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 713	. . 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 3952	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ (𝐽 fLim 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  {csn 4589  ∪ cuni 4871  ‘cfv 6511  (class class class)co 7387  Topctop 22780  TopOnctopon 22797  clsccl 22905  neicnei 22984  Filcfil 23732   fLim cflim 23821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  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 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-fbas 21261  df-top 22781  df-topon 22798  df-cld 22906  df-ntr 22907  df-cls 22908  df-nei 22985  df-fil 23733  df-flim 23826

This theorem is referenced by:  tsmscls  24025