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Theorem flimsncls 22011
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.
StepHypRef Expression
1 flimtop 21990 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2 eqid 2760 . . . . . . . 8 𝐽 = 𝐽
32flimelbas 21993 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 𝐽)
43snssd 4485 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → {𝐴} ⊆ 𝐽)
52clsss3 21085 . . . . . 6 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽) → ((cls‘𝐽)‘{𝐴}) ⊆ 𝐽)
61, 4, 5syl2anc 696 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ 𝐽)
76sselda 3744 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 𝐽)
8 simpll 807 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝐴 ∈ (𝐽 fLim 𝐹))
98, 1syl 17 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝐽 ∈ Top)
10 simprl 811 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝑦𝐽)
111adantr 472 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐽 ∈ Top)
124adantr 472 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → {𝐴} ⊆ 𝐽)
13 simpr 479 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 ∈ ((cls‘𝐽)‘{𝐴}))
1411, 12, 133jca 1123 . . . . . . . . 9 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → (𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑥 ∈ ((cls‘𝐽)‘{𝐴})))
152clsndisj 21101 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → (𝑦 ∩ {𝐴}) ≠ ∅)
16 disjsn 4390 . . . . . . . . . . 11 ((𝑦 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝑦)
1716necon2abii 2982 . . . . . . . . . 10 (𝐴𝑦 ↔ (𝑦 ∩ {𝐴}) ≠ ∅)
1815, 17sylibr 224 . . . . . . . . 9 (((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝐴𝑦)
1914, 18sylan 489 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝐴𝑦)
20 opnneip 21145 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑦𝐽𝐴𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
219, 10, 19, 20syl3anc 1477 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
22 flimnei 21992 . . . . . . 7 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦𝐹)
238, 21, 22syl2anc 696 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝑦𝐹)
2423expr 644 . . . . 5 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ 𝑦𝐽) → (𝑥𝑦𝑦𝐹))
2524ralrimiva 3104 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → ∀𝑦𝐽 (𝑥𝑦𝑦𝐹))
262toptopon 20944 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
2711, 26sylib 208 . . . . 5 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐽 ∈ (TopOn‘ 𝐽))
282flimfil 21994 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
2928adantr 472 . . . . 5 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐹 ∈ (Fil‘ 𝐽))
30 flimopn 22000 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹 ∈ (Fil‘ 𝐽)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 𝐽 ∧ ∀𝑦𝐽 (𝑥𝑦𝑦𝐹))))
3127, 29, 30syl2anc 696 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 𝐽 ∧ ∀𝑦𝐽 (𝑥𝑦𝑦𝐹))))
327, 25, 31mpbir2and 995 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 ∈ (𝐽 fLim 𝐹))
3332ex 449 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑥 ∈ ((cls‘𝐽)‘{𝐴}) → 𝑥 ∈ (𝐽 fLim 𝐹)))
3433ssrdv 3750 1 (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ (𝐽 fLim 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wcel 2139  wne 2932  wral 3050  cin 3714  wss 3715  c0 4058  {csn 4321   cuni 4588  cfv 6049  (class class class)co 6814  Topctop 20920  TopOnctopon 20937  clsccl 21044  neicnei 21123  Filcfil 21870   fLim cflim 21959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-fbas 19965  df-top 20921  df-topon 20938  df-cld 21045  df-ntr 21046  df-cls 21047  df-nei 21124  df-fil 21871  df-flim 21964
This theorem is referenced by:  tsmscls  22162
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