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Theorem flimsncls 23374
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 23353 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2 eqid 2731 . . . . . . . 8 𝐽 = 𝐽
32flimelbas 23356 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 𝐽)
43snssd 4774 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → {𝐴} ⊆ 𝐽)
52clsss3 22447 . . . . . 6 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽) → ((cls‘𝐽)‘{𝐴}) ⊆ 𝐽)
61, 4, 5syl2anc 584 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ 𝐽)
76sselda 3947 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 𝐽)
8 simpll 765 . . . . . . 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 769 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝑦𝐽)
111adantr 481 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐽 ∈ Top)
124adantr 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}))
1411, 12, 133jca 1128 . . . . . . . . 9 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → (𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑥 ∈ ((cls‘𝐽)‘{𝐴})))
152clsndisj 22463 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → (𝑦 ∩ {𝐴}) ≠ ∅)
16 disjsn 4677 . . . . . . . . . . 11 ((𝑦 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝑦)
1716necon2abii 2990 . . . . . . . . . 10 (𝐴𝑦 ↔ (𝑦 ∩ {𝐴}) ≠ ∅)
1815, 17sylibr 233 . . . . . . . . 9 (((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝐴𝑦)
1914, 18sylan 580 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝐴𝑦)
20 opnneip 22507 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑦𝐽𝐴𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
219, 10, 19, 20syl3anc 1371 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
22 flimnei 23355 . . . . . . 7 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦𝐹)
238, 21, 22syl2anc 584 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝑦𝐹)
2423expr 457 . . . . 5 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ 𝑦𝐽) → (𝑥𝑦𝑦𝐹))
2524ralrimiva 3139 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → ∀𝑦𝐽 (𝑥𝑦𝑦𝐹))
26 toptopon2 22304 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
2711, 26sylib 217 . . . . 5 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐽 ∈ (TopOn‘ 𝐽))
282flimfil 23357 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
2928adantr 481 . . . . 5 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐹 ∈ (Fil‘ 𝐽))
30 flimopn 23363 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹 ∈ (Fil‘ 𝐽)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 𝐽 ∧ ∀𝑦𝐽 (𝑥𝑦𝑦𝐹))))
3127, 29, 30syl2anc 584 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 𝐽 ∧ ∀𝑦𝐽 (𝑥𝑦𝑦𝐹))))
327, 25, 31mpbir2and 711 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 ∈ (𝐽 fLim 𝐹))
3332ex 413 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑥 ∈ ((cls‘𝐽)‘{𝐴}) → 𝑥 ∈ (𝐽 fLim 𝐹)))
3433ssrdv 3953 1 (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ (𝐽 fLim 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087  wcel 2106  wne 2939  wral 3060  cin 3912  wss 3913  c0 4287  {csn 4591   cuni 4870  cfv 6501  (class class class)co 7362  Topctop 22279  TopOnctopon 22296  clsccl 22406  neicnei 22485  Filcfil 23233   fLim cflim 23322
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  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 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-fbas 20830  df-top 22280  df-topon 22297  df-cld 22407  df-ntr 22408  df-cls 22409  df-nei 22486  df-fil 23234  df-flim 23327
This theorem is referenced by:  tsmscls  23526
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