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Theorem flimtopon 22506
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 22501 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2 istopon 21448 . . . 4 (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝐽))
32baib 536 . . 3 (𝐽 ∈ Top → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = 𝐽))
41, 3syl 17 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = 𝐽))
5 eqid 2818 . . . . 5 𝐽 = 𝐽
65flimfil 22505 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
7 fveq2 6663 . . . . 5 (𝑋 = 𝐽 → (Fil‘𝑋) = (Fil‘ 𝐽))
87eleq2d 2895 . . . 4 (𝑋 = 𝐽 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘ 𝐽)))
96, 8syl5ibrcom 248 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑋 = 𝐽𝐹 ∈ (Fil‘𝑋)))
10 filunibas 22417 . . . . 5 (𝐹 ∈ (Fil‘ 𝐽) → 𝐹 = 𝐽)
116, 10syl 17 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 = 𝐽)
12 filunibas 22417 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
1312eqeq1d 2820 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ( 𝐹 = 𝐽𝑋 = 𝐽))
1411, 13syl5ibcom 246 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐹 ∈ (Fil‘𝑋) → 𝑋 = 𝐽))
159, 14impbid 213 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑋 = 𝐽𝐹 ∈ (Fil‘𝑋)))
164, 15bitrd 280 1 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105   cuni 4830  cfv 6348  (class class class)co 7145  Topctop 21429  TopOnctopon 21446  Filcfil 22381   fLim cflim 22470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-fbas 20470  df-top 21430  df-topon 21447  df-nei 21634  df-fil 22382  df-flim 22475
This theorem is referenced by:  fclsfnflim  22563
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