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Theorem flimtopon 23456
Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
flimtopon (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))

Proof of Theorem flimtopon
StepHypRef Expression
1 flimtop 23451 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2 istopon 22396 . . . 4 (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝐽))
32baib 537 . . 3 (𝐽 ∈ Top → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = 𝐽))
41, 3syl 17 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = 𝐽))
5 eqid 2733 . . . . 5 𝐽 = 𝐽
65flimfil 23455 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
7 fveq2 6888 . . . . 5 (𝑋 = 𝐽 → (Fil‘𝑋) = (Fil‘ 𝐽))
87eleq2d 2820 . . . 4 (𝑋 = 𝐽 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘ 𝐽)))
96, 8syl5ibrcom 246 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑋 = 𝐽𝐹 ∈ (Fil‘𝑋)))
10 filunibas 23367 . . . . 5 (𝐹 ∈ (Fil‘ 𝐽) → 𝐹 = 𝐽)
116, 10syl 17 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 = 𝐽)
12 filunibas 23367 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
1312eqeq1d 2735 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ( 𝐹 = 𝐽𝑋 = 𝐽))
1411, 13syl5ibcom 244 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐹 ∈ (Fil‘𝑋) → 𝑋 = 𝐽))
159, 14impbid 211 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑋 = 𝐽𝐹 ∈ (Fil‘𝑋)))
164, 15bitrd 279 1 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107   cuni 4907  cfv 6540  (class class class)co 7404  Topctop 22377  TopOnctopon 22394  Filcfil 23331   fLim cflim 23420
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-fbas 20926  df-top 22378  df-topon 22395  df-nei 22584  df-fil 23332  df-flim 23425
This theorem is referenced by:  fclsfnflim  23513
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