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Theorem flimsncls 21730
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 21709 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2 eqid 2621 . . . . . . . 8 𝐽 = 𝐽
32flimelbas 21712 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 𝐽)
43snssd 4316 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → {𝐴} ⊆ 𝐽)
52clsss3 20803 . . . . . 6 ((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽) → ((cls‘𝐽)‘{𝐴}) ⊆ 𝐽)
61, 4, 5syl2anc 692 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ 𝐽)
76sselda 3588 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 𝐽)
8 simpll 789 . . . . . . 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 793 . . . . . . . 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 477 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 ∈ ((cls‘𝐽)‘{𝐴}))
1411, 12, 133jca 1240 . . . . . . . . 9 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → (𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑥 ∈ ((cls‘𝐽)‘{𝐴})))
152clsndisj 20819 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → (𝑦 ∩ {𝐴}) ≠ ∅)
16 disjsn 4223 . . . . . . . . . . 11 ((𝑦 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴𝑦)
1716necon2abii 2840 . . . . . . . . . 10 (𝐴𝑦 ↔ (𝑦 ∩ {𝐴}) ≠ ∅)
1815, 17sylibr 224 . . . . . . . . 9 (((𝐽 ∈ Top ∧ {𝐴} ⊆ 𝐽𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝐴𝑦)
1914, 18sylan 488 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝐴𝑦)
20 opnneip 20863 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑦𝐽𝐴𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
219, 10, 19, 20syl3anc 1323 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
22 flimnei 21711 . . . . . . 7 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑦𝐹)
238, 21, 22syl2anc 692 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ (𝑦𝐽𝑥𝑦)) → 𝑦𝐹)
2423expr 642 . . . . 5 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) ∧ 𝑦𝐽) → (𝑥𝑦𝑦𝐹))
2524ralrimiva 2962 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → ∀𝑦𝐽 (𝑥𝑦𝑦𝐹))
262toptopon 20662 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
2711, 26sylib 208 . . . . 5 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐽 ∈ (TopOn‘ 𝐽))
282flimfil 21713 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
2928adantr 481 . . . . 5 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝐹 ∈ (Fil‘ 𝐽))
30 flimopn 21719 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐹 ∈ (Fil‘ 𝐽)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 𝐽 ∧ ∀𝑦𝐽 (𝑥𝑦𝑦𝐹))))
3127, 29, 30syl2anc 692 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 𝐽 ∧ ∀𝑦𝐽 (𝑥𝑦𝑦𝐹))))
327, 25, 31mpbir2and 956 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ ((cls‘𝐽)‘{𝐴})) → 𝑥 ∈ (𝐽 fLim 𝐹))
3332ex 450 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑥 ∈ ((cls‘𝐽)‘{𝐴}) → 𝑥 ∈ (𝐽 fLim 𝐹)))
3433ssrdv 3594 1 (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ (𝐽 fLim 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wcel 1987  wne 2790  wral 2908  cin 3559  wss 3560  c0 3897  {csn 4155   cuni 4409  cfv 5857  (class class class)co 6615  Topctop 20638  TopOnctopon 20655  clsccl 20762  neicnei 20841  Filcfil 21589   fLim cflim 21678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-fbas 19683  df-top 20639  df-topon 20656  df-cld 20763  df-ntr 20764  df-cls 20765  df-nei 20842  df-fil 21590  df-flim 21683
This theorem is referenced by:  tsmscls  21881
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