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Theorem flimsncls 21600
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 21579 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2 eqid 2610 . . . . . . . 8 𝐽 = 𝐽
32flimelbas 21582 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 𝐽)
43snssd 4281 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → {𝐴} ⊆ 𝐽)
52clsss3 20673 . . . . . 6 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽) → ((cls‘𝐽)‘{𝐴}) ⊆ 𝐽)
61, 4, 5syl2anc 691 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ 𝐽)
76sselda 3568 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 𝐽)
8 simpll 786 . . . . . . 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 790 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝑦𝐽)
111adantr 480 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐽 ∈ Top)
124adantr 480 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → {𝐴} ⊆ 𝐽)
13 simpr 476 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 ∈ ((cls‘𝐽)‘{𝐴}))
1411, 12, 133jca 1235 . . . . . . . . 9 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → (𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑥 ∈ ((cls‘𝐽)‘{𝐴})))
152clsndisj 20689 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → (𝑦 ∩ {𝐴}) ≠ ∅)
16 disjsn 4192 . . . . . . . . . . 11 ((𝑦 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝑦)
1716necon2abii 2832 . . . . . . . . . 10 (𝐴𝑦 ↔ (𝑦 ∩ {𝐴}) ≠ ∅)
1815, 17sylibr 223 . . . . . . . . 9 (((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝐴𝑦)
1914, 18sylan 487 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝐴𝑦)
20 opnneip 20733 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑦𝐽𝐴𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
219, 10, 19, 20syl3anc 1318 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
22 flimnei 21581 . . . . . . 7 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦𝐹)
238, 21, 22syl2anc 691 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝑦𝐹)
2423expr 641 . . . . 5 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ 𝑦𝐽) → (𝑥𝑦𝑦𝐹))
2524ralrimiva 2949 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → ∀𝑦𝐽 (𝑥𝑦𝑦𝐹))
262toptopon 20548 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
2711, 26sylib 207 . . . . 5 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐽 ∈ (TopOn‘ 𝐽))
282flimfil 21583 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
2928adantr 480 . . . . 5 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐹 ∈ (Fil‘ 𝐽))
30 flimopn 21589 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹 ∈ (Fil‘ 𝐽)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 𝐽 ∧ ∀𝑦𝐽 (𝑥𝑦𝑦𝐹))))
3127, 29, 30syl2anc 691 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 𝐽 ∧ ∀𝑦𝐽 (𝑥𝑦𝑦𝐹))))
327, 25, 31mpbir2and 959 . . 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 3574 1 (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ (𝐽 fLim 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031  wcel 1977  wne 2780  wral 2896  cin 3539  wss 3540  c0 3874  {csn 4125   cuni 4372  cfv 5804  (class class class)co 6549  Topctop 20517  TopOnctopon 20518  clsccl 20632  neicnei 20711  Filcfil 21459   fLim cflim 21548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-fbas 19564  df-top 20521  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-fil 21460  df-flim 21553
This theorem is referenced by:  tsmscls  21751
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