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Theorem flimsncls 23490
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) β†’ ((clsβ€˜π½)β€˜{𝐴}) βŠ† (𝐽 fLim 𝐹))

Proof of Theorem flimsncls
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 flimtop 23469 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐽 ∈ Top)
2 eqid 2733 . . . . . . . 8 βˆͺ 𝐽 = βˆͺ 𝐽
32flimelbas 23472 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐴 ∈ βˆͺ 𝐽)
43snssd 4813 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ {𝐴} βŠ† βˆͺ 𝐽)
52clsss3 22563 . . . . . 6 ((𝐽 ∈ Top ∧ {𝐴} βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜{𝐴}) βŠ† βˆͺ 𝐽)
61, 4, 5syl2anc 585 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ ((clsβ€˜π½)β€˜{𝐴}) βŠ† βˆͺ 𝐽)
76sselda 3983 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ π‘₯ ∈ βˆͺ 𝐽)
8 simpll 766 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ π‘₯ ∈ 𝑦)) β†’ 𝐴 ∈ (𝐽 fLim 𝐹))
98, 1syl 17 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ π‘₯ ∈ 𝑦)) β†’ 𝐽 ∈ Top)
10 simprl 770 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ π‘₯ ∈ 𝑦)) β†’ 𝑦 ∈ 𝐽)
111adantr 482 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ 𝐽 ∈ Top)
124adantr 482 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ {𝐴} βŠ† βˆͺ 𝐽)
13 simpr 486 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴}))
1411, 12, 133jca 1129 . . . . . . . . 9 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ (𝐽 ∈ Top ∧ {𝐴} βŠ† βˆͺ 𝐽 ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})))
152clsndisj 22579 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ {𝐴} βŠ† βˆͺ 𝐽 ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ π‘₯ ∈ 𝑦)) β†’ (𝑦 ∩ {𝐴}) β‰  βˆ…)
16 disjsn 4716 . . . . . . . . . . 11 ((𝑦 ∩ {𝐴}) = βˆ… ↔ Β¬ 𝐴 ∈ 𝑦)
1716necon2abii 2992 . . . . . . . . . 10 (𝐴 ∈ 𝑦 ↔ (𝑦 ∩ {𝐴}) β‰  βˆ…)
1815, 17sylibr 233 . . . . . . . . 9 (((𝐽 ∈ Top ∧ {𝐴} βŠ† βˆͺ 𝐽 ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ π‘₯ ∈ 𝑦)) β†’ 𝐴 ∈ 𝑦)
1914, 18sylan 581 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ π‘₯ ∈ 𝑦)) β†’ 𝐴 ∈ 𝑦)
20 opnneip 22623 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) β†’ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}))
219, 10, 19, 20syl3anc 1372 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ π‘₯ ∈ 𝑦)) β†’ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}))
22 flimnei 23471 . . . . . . 7 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝑦 ∈ 𝐹)
238, 21, 22syl2anc 585 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ π‘₯ ∈ 𝑦)) β†’ 𝑦 ∈ 𝐹)
2423expr 458 . . . . 5 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) ∧ 𝑦 ∈ 𝐽) β†’ (π‘₯ ∈ 𝑦 β†’ 𝑦 ∈ 𝐹))
2524ralrimiva 3147 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ βˆ€π‘¦ ∈ 𝐽 (π‘₯ ∈ 𝑦 β†’ 𝑦 ∈ 𝐹))
26 toptopon2 22420 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
2711, 26sylib 217 . . . . 5 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
282flimfil 23473 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐽))
2928adantr 482 . . . . 5 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐽))
30 flimopn 23479 . . . . 5 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ 𝐹 ∈ (Filβ€˜βˆͺ 𝐽)) β†’ (π‘₯ ∈ (𝐽 fLim 𝐹) ↔ (π‘₯ ∈ βˆͺ 𝐽 ∧ βˆ€π‘¦ ∈ 𝐽 (π‘₯ ∈ 𝑦 β†’ 𝑦 ∈ 𝐹))))
3127, 29, 30syl2anc 585 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ (π‘₯ ∈ (𝐽 fLim 𝐹) ↔ (π‘₯ ∈ βˆͺ 𝐽 ∧ βˆ€π‘¦ ∈ 𝐽 (π‘₯ ∈ 𝑦 β†’ 𝑦 ∈ 𝐹))))
327, 25, 31mpbir2and 712 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ π‘₯ ∈ (𝐽 fLim 𝐹))
3332ex 414 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴}) β†’ π‘₯ ∈ (𝐽 fLim 𝐹)))
3433ssrdv 3989 1 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ ((clsβ€˜π½)β€˜{𝐴}) βŠ† (𝐽 fLim 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  (class class class)co 7409  Topctop 22395  TopOnctopon 22412  clsccl 22522  neicnei 22601  Filcfil 23349   fLim cflim 23438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-fbas 20941  df-top 22396  df-topon 22413  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-fil 23350  df-flim 23443
This theorem is referenced by:  tsmscls  23642
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