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Theorem elflim 23995
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 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))

Proof of Theorem elflim
StepHypRef Expression
1 topontop 22935 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21adantr 480 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐽 ∈ Top)
3 fvssunirn 6940 . . . . 5 (Fil‘𝑋) ⊆ ran Fil
43sseli 3991 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ran Fil)
54adantl 481 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ran Fil)
6 filsspw 23875 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
76adantl 481 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
8 toponuni 22936 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
98adantr 480 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝑋 = 𝐽)
109pweqd 4622 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝒫 𝑋 = 𝒫 𝐽)
117, 10sseqtrd 4036 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ⊆ 𝒫 𝐽)
12 eqid 2735 . . . . 5 𝐽 = 𝐽
1312elflim2 23988 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝐽) ∧ (𝐴 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
1413baib 535 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝐽) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
152, 5, 11, 14syl3anc 1370 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
169eleq2d 2825 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴𝑋𝐴 𝐽))
1716anbi1d 631 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹) ↔ (𝐴 𝐽 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
1815, 17bitr4d 282 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wss 3963  𝒫 cpw 4605  {csn 4631   cuni 4912  ran crn 5690  cfv 6563  (class class class)co 7431  Topctop 22915  TopOnctopon 22932  neicnei 23121  Filcfil 23869   fLim cflim 23958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-top 22916  df-topon 22933  df-fil 23870  df-flim 23963
This theorem is referenced by:  flimss2  23996  flimss1  23997  neiflim  23998  flimopn  23999  hausflim  24005  flimclslem  24008  flfnei  24015  fclsfnflim  24051
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