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Theorem elflim 23695
Description: The predicate "is a limit point of a filter." (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
elflim ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
Proof of Theorem elflim
StepHypRef Expression
1 topontop 22635 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
21adantr 481 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ 𝐽 ∈ Top)
3 fvssunirn 6924 . . . . 5 (Filβ€˜π‘‹) βŠ† βˆͺ ran Fil
43sseli 3978 . . . 4 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 ∈ βˆͺ ran Fil)
54adantl 482 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ 𝐹 ∈ βˆͺ ran Fil)
6 filsspw 23575 . . . . 5 (𝐹 ∈ (Filβ€˜π‘‹) β†’ 𝐹 βŠ† 𝒫 𝑋)
76adantl 482 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ 𝐹 βŠ† 𝒫 𝑋)
8 toponuni 22636 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
98adantr 481 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ 𝑋 = βˆͺ 𝐽)
109pweqd 4619 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ 𝒫 𝑋 = 𝒫 βˆͺ 𝐽)
117, 10sseqtrd 4022 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ 𝐹 βŠ† 𝒫 βˆͺ 𝐽)
12 eqid 2732 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
1312elflim2 23688 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil ∧ 𝐹 βŠ† 𝒫 βˆͺ 𝐽) ∧ (𝐴 ∈ βˆͺ 𝐽 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
1413baib 536 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil ∧ 𝐹 βŠ† 𝒫 βˆͺ 𝐽) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ βˆͺ 𝐽 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
152, 5, 11, 14syl3anc 1371 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ βˆͺ 𝐽 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
169eleq2d 2819 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ βˆͺ 𝐽))
1716anbi1d 630 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ ((𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹) ↔ (𝐴 ∈ βˆͺ 𝐽 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
1815, 17bitr4d 281 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βŠ† wss 3948  π’« cpw 4602  {csn 4628  βˆͺ cuni 4908  ran crn 5677  β€˜cfv 6543  (class class class)co 7411  Topctop 22615  TopOnctopon 22632  neicnei 22821  Filcfil 23569   fLim cflim 23658
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-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-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-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-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 21141  df-top 22616  df-topon 22633  df-fil 23570  df-flim 23663
This theorem is referenced by:  flimss2  23696  flimss1  23697  neiflim  23698  flimopn  23699  hausflim  23705  flimclslem  23708  flfnei  23715  fclsfnflim  23751
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