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Theorem flimtopon 22579
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 22574 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2 istopon 21521 . . . 4 (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝐽))
32baib 539 . . 3 (𝐽 ∈ Top → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = 𝐽))
41, 3syl 17 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = 𝐽))
5 eqid 2801 . . . . 5 𝐽 = 𝐽
65flimfil 22578 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
7 fveq2 6649 . . . . 5 (𝑋 = 𝐽 → (Fil‘𝑋) = (Fil‘ 𝐽))
87eleq2d 2878 . . . 4 (𝑋 = 𝐽 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘ 𝐽)))
96, 8syl5ibrcom 250 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑋 = 𝐽𝐹 ∈ (Fil‘𝑋)))
10 filunibas 22490 . . . . 5 (𝐹 ∈ (Fil‘ 𝐽) → 𝐹 = 𝐽)
116, 10syl 17 . . . 4 (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 = 𝐽)
12 filunibas 22490 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
1312eqeq1d 2803 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ( 𝐹 = 𝐽𝑋 = 𝐽))
1411, 13syl5ibcom 248 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐹 ∈ (Fil‘𝑋) → 𝑋 = 𝐽))
159, 14impbid 215 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑋 = 𝐽𝐹 ∈ (Fil‘𝑋)))
164, 15bitrd 282 1 (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2112   cuni 4803  cfv 6328  (class class class)co 7139  Topctop 21502  TopOnctopon 21519  Filcfil 22454   fLim cflim 22543
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-fbas 20092  df-top 21503  df-topon 21520  df-nei 21707  df-fil 22455  df-flim 22548
This theorem is referenced by:  fclsfnflim  22636
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