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Theorem flimsncls 22209
 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 22188 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2 eqid 2778 . . . . . . . 8 𝐽 = 𝐽
32flimelbas 22191 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 𝐽)
43snssd 4573 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → {𝐴} ⊆ 𝐽)
52clsss3 21282 . . . . . 6 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽) → ((cls‘𝐽)‘{𝐴}) ⊆ 𝐽)
61, 4, 5syl2anc 579 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ 𝐽)
76sselda 3821 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 𝐽)
8 simpll 757 . . . . . . 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 761 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝑦𝐽)
111adantr 474 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐽 ∈ Top)
124adantr 474 . . . . . . . . . 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 1119 . . . . . . . . 9 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → (𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑥 ∈ ((cls‘𝐽)‘{𝐴})))
152clsndisj 21298 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → (𝑦 ∩ {𝐴}) ≠ ∅)
16 disjsn 4478 . . . . . . . . . . 11 ((𝑦 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝑦)
1716necon2abii 3019 . . . . . . . . . 10 (𝐴𝑦 ↔ (𝑦 ∩ {𝐴}) ≠ ∅)
1815, 17sylibr 226 . . . . . . . . 9 (((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝐴𝑦)
1914, 18sylan 575 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝐴𝑦)
20 opnneip 21342 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑦𝐽𝐴𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
219, 10, 19, 20syl3anc 1439 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
22 flimnei 22190 . . . . . . 7 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦𝐹)
238, 21, 22syl2anc 579 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝑦𝐹)
2423expr 450 . . . . 5 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ 𝑦𝐽) → (𝑥𝑦𝑦𝐹))
2524ralrimiva 3148 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → ∀𝑦𝐽 (𝑥𝑦𝑦𝐹))
26 toptopon2 21141 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
2711, 26sylib 210 . . . . 5 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐽 ∈ (TopOn‘ 𝐽))
282flimfil 22192 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
2928adantr 474 . . . . 5 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐹 ∈ (Fil‘ 𝐽))
30 flimopn 22198 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹 ∈ (Fil‘ 𝐽)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 𝐽 ∧ ∀𝑦𝐽 (𝑥𝑦𝑦𝐹))))
3127, 29, 30syl2anc 579 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 𝐽 ∧ ∀𝑦𝐽 (𝑥𝑦𝑦𝐹))))
327, 25, 31mpbir2and 703 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 ∈ (𝐽 fLim 𝐹))
3332ex 403 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑥 ∈ ((cls‘𝐽)‘{𝐴}) → 𝑥 ∈ (𝐽 fLim 𝐹)))
3433ssrdv 3827 1 (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ (𝐽 fLim 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141  {csn 4398  ∪ cuni 4673  ‘cfv 6137  (class class class)co 6924  Topctop 21116  TopOnctopon 21133  clsccl 21241  neicnei 21320  Filcfil 22068   fLim cflim 22157 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-iin 4758  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-fbas 20150  df-top 21117  df-topon 21134  df-cld 21242  df-ntr 21243  df-cls 21244  df-nei 21321  df-fil 22069  df-flim 22162 This theorem is referenced by:  tsmscls  22360
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