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Theorem flimval 23298
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 22239 . . . 4 (𝐽 ∈ Top → 𝑋𝐽)
32adantr 481 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → 𝑋𝐽)
4 rabexg 5286 . . 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 4877 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
87, 1eqtr4di 2794 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝑋)
96fveq2d 6843 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → (nei‘𝑗) = (nei‘𝐽))
109fveq1d 6841 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
11 simpr 485 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑓 = 𝐹)
1210, 11sseq12d 3975 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ↔ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹))
138pweqd 4575 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝒫 𝑗 = 𝒫 𝑋)
1411, 13sseq12d 3975 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → (𝑓 ⊆ 𝒫 𝑗𝐹 ⊆ 𝒫 𝑋))
1512, 14anbi12d 631 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → ((((nei‘𝑗)‘{𝑥}) ⊆ 𝑓𝑓 ⊆ 𝒫 𝑗) ↔ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹𝐹 ⊆ 𝒫 𝑋)))
168, 15rabeqbidv 3422 . . 3 ((𝑗 = 𝐽𝑓 = 𝐹) → {𝑥 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓𝑓 ⊆ 𝒫 𝑗)} = {𝑥𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹𝐹 ⊆ 𝒫 𝑋)})
17 df-flim 23274 . . 3 fLim = (𝑗 ∈ Top, 𝑓 ran Fil ↦ {𝑥 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓𝑓 ⊆ 𝒫 𝑗)})
1816, 17ovmpoga 7505 . 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 1462 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 1541  wcel 2106  {crab 3405  Vcvv 3443  wss 3908  𝒫 cpw 4558  {csn 4584   cuni 4863  ran crn 5632  cfv 6493  (class class class)co 7353  Topctop 22226  neicnei 22432  Filcfil 23180   fLim cflim 23269
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-top 22227  df-flim 23274
This theorem is referenced by:  elflim2  23299
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