Theorem flimsncls 23976

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 23955	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2		eqid 2740	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	flimelbas 23958	. . . . . . 7 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 ∈ ∪ 𝐽)
4	3	snssd 4725	. . . . . 6 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → {𝐴} ⊆ ∪ 𝐽)
5	2	clsss3 23049	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽) → ((cls‘𝐽)‘{𝐴}) ⊆ ∪ 𝐽)
6	1, 4, 5	syl2anc 590	. . . . 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 772	. . . . . . 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 776	. . . . . . . 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 1134	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → (𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})))
15	2	clsndisj 23065	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → (𝑦 ∩ {𝐴}) ≠ ∅)
16		disjsn 4650	. . . . . . . . . . 11 ⊢ ((𝑦 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝑦)
17	16	necon2abii 2985	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑦 ↔ (𝑦 ∩ {𝐴}) ≠ ∅)
18	15, 17	sylibr 235	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝐴 ∈ 𝑦)
19	14, 18	sylan 586	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝐴 ∈ 𝑦)
20		opnneip 23109	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
21	9, 10, 19, 20	syl3anc 1379	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
22		flimnei 23957	. . . . . . 7 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦 ∈ 𝐹)
23	8, 21, 22	syl2anc 590	. . . . . 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 3132	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → ∀𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))
26		toptopon2 22908	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
27	11, 26	sylib 219	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐽 ∈ (TopOn‘∪ 𝐽))
28	2	flimfil 23959	. . . . . 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 23965	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐹 ∈ (Fil‘∪ 𝐽)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹))))
31	27, 29, 30	syl2anc 590	. . . 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 719	. . 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 3928	1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ (𝐽 fLim 𝐹))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268  {csn 4562  ∪ cuni 4845  ‘cfv 6492  (class class class)co 7363  Topctop 22883  TopOnctopon 22900  clsccl 23008  neicnei 23087  Filcfil 23835   fLim cflim 23924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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 7366  df-oprab 7367  df-mpo 7368  df-fbas 21351  df-top 22884  df-topon 22901  df-cld 23009  df-ntr 23010  df-cls 23011  df-nei 23088  df-fil 23836  df-flim 23929

This theorem is referenced by:  tsmscls  24128