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Theorem flimsncls 23497
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 23476 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐽 ∈ Top)
2 eqid 2732 . . . . . . . 8 βˆͺ 𝐽 = βˆͺ 𝐽
32flimelbas 23479 . . . . . . 7 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐴 ∈ βˆͺ 𝐽)
43snssd 4812 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ {𝐴} βŠ† βˆͺ 𝐽)
52clsss3 22570 . . . . . 6 ((𝐽 ∈ Top ∧ {𝐴} βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜{𝐴}) βŠ† βˆͺ 𝐽)
61, 4, 5syl2anc 584 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ ((clsβ€˜π½)β€˜{𝐴}) βŠ† βˆͺ 𝐽)
76sselda 3982 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ π‘₯ ∈ βˆͺ 𝐽)
8 simpll 765 . . . . . . 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 769 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ π‘₯ ∈ 𝑦)) β†’ 𝑦 ∈ 𝐽)
111adantr 481 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ 𝐽 ∈ Top)
124adantr 481 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ {𝐴} βŠ† βˆͺ 𝐽)
13 simpr 485 . . . . . . . . . 10 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴}))
1411, 12, 133jca 1128 . . . . . . . . 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 22586 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ {𝐴} βŠ† βˆͺ 𝐽 ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ π‘₯ ∈ 𝑦)) β†’ (𝑦 ∩ {𝐴}) β‰  βˆ…)
16 disjsn 4715 . . . . . . . . . . 11 ((𝑦 ∩ {𝐴}) = βˆ… ↔ Β¬ 𝐴 ∈ 𝑦)
1716necon2abii 2991 . . . . . . . . . 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 580 . . . . . . . 8 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ π‘₯ ∈ 𝑦)) β†’ 𝐴 ∈ 𝑦)
20 opnneip 22630 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) β†’ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}))
219, 10, 19, 20syl3anc 1371 . . . . . . 7 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ π‘₯ ∈ 𝑦)) β†’ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}))
22 flimnei 23478 . . . . . . 7 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝑦 ∈ 𝐹)
238, 21, 22syl2anc 584 . . . . . 6 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) ∧ (𝑦 ∈ 𝐽 ∧ π‘₯ ∈ 𝑦)) β†’ 𝑦 ∈ 𝐹)
2423expr 457 . . . . 5 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) ∧ 𝑦 ∈ 𝐽) β†’ (π‘₯ ∈ 𝑦 β†’ 𝑦 ∈ 𝐹))
2524ralrimiva 3146 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ βˆ€π‘¦ ∈ 𝐽 (π‘₯ ∈ 𝑦 β†’ 𝑦 ∈ 𝐹))
26 toptopon2 22427 . . . . . 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 23480 . . . . . 6 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐽))
2928adantr 481 . . . . 5 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐽))
30 flimopn 23486 . . . . 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 584 . . . 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 711 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴})) β†’ π‘₯ ∈ (𝐽 fLim 𝐹))
3332ex 413 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜{𝐴}) β†’ π‘₯ ∈ (𝐽 fLim 𝐹)))
3433ssrdv 3988 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 396   ∧ w3a 1087   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  βˆͺ cuni 4908  β€˜cfv 6543  (class class class)co 7411  Topctop 22402  TopOnctopon 22419  clsccl 22529  neicnei 22608  Filcfil 23356   fLim cflim 23445
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 20947  df-top 22403  df-topon 22420  df-cld 22530  df-ntr 22531  df-cls 22532  df-nei 22609  df-fil 23357  df-flim 23450
This theorem is referenced by:  tsmscls  23649
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