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Theorem flimval 22499
Description: The set of limit points of a filter. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
flimval.1 𝑋 = 𝐽
Assertion
Ref Expression
flimval ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐽 fLim 𝐹) = {𝑥𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹𝐹 ⊆ 𝒫 𝑋)})
Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝑋

Proof of Theorem flimval
Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 flimval.1 . . . . 5 𝑋 = 𝐽
21topopn 21442 . . . 4 (𝐽 ∈ Top → 𝑋𝐽)
32adantr 481 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → 𝑋𝐽)
4 rabexg 5225 . . 3 (𝑋𝐽 → {𝑥𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹𝐹 ⊆ 𝒫 𝑋)} ∈ V)
53, 4syl 17 . 2 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → {𝑥𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹𝐹 ⊆ 𝒫 𝑋)} ∈ V)
6 simpl 483 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
76unieqd 4840 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
87, 1syl6eqr 2871 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝑋)
96fveq2d 6667 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → (nei‘𝑗) = (nei‘𝐽))
109fveq1d 6665 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
11 simpr 485 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑓 = 𝐹)
1210, 11sseq12d 3997 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ↔ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹))
138pweqd 4540 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝒫 𝑗 = 𝒫 𝑋)
1411, 13sseq12d 3997 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → (𝑓 ⊆ 𝒫 𝑗𝐹 ⊆ 𝒫 𝑋))
1512, 14anbi12d 630 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → ((((nei‘𝑗)‘{𝑥}) ⊆ 𝑓𝑓 ⊆ 𝒫 𝑗) ↔ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹𝐹 ⊆ 𝒫 𝑋)))
168, 15rabeqbidv 3483 . . 3 ((𝑗 = 𝐽𝑓 = 𝐹) → {𝑥 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓𝑓 ⊆ 𝒫 𝑗)} = {𝑥𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹𝐹 ⊆ 𝒫 𝑋)})
17 df-flim 22475 . . 3 fLim = (𝑗 ∈ Top, 𝑓 ran Fil ↦ {𝑥 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓𝑓 ⊆ 𝒫 𝑗)})
1816, 17ovmpoga 7293 . 2 ((𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ {𝑥𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹𝐹 ⊆ 𝒫 𝑋)} ∈ V) → (𝐽 fLim 𝐹) = {𝑥𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹𝐹 ⊆ 𝒫 𝑋)})
195, 18mpd3an3 1453 1 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐽 fLim 𝐹) = {𝑥𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹𝐹 ⊆ 𝒫 𝑋)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  wss 3933  𝒫 cpw 4535  {csn 4557   cuni 4830  ran crn 5549  cfv 6348  (class class class)co 7145  Topctop 21429  neicnei 21633  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-sep 5194  ax-nul 5201  ax-pr 5320
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-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  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-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-top 21430  df-flim 22475
This theorem is referenced by:  elflim2  22500
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