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Theorem flimval 23971
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 22912 . . . 4 (𝐽 ∈ Top → 𝑋𝐽)
32adantr 480 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → 𝑋𝐽)
4 rabexg 5337 . . 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 482 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
76unieqd 4920 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
87, 1eqtr4di 2795 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝑋)
96fveq2d 6910 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → (nei‘𝑗) = (nei‘𝐽))
109fveq1d 6908 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
11 simpr 484 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑓 = 𝐹)
1210, 11sseq12d 4017 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ↔ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹))
138pweqd 4617 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝒫 𝑗 = 𝒫 𝑋)
1411, 13sseq12d 4017 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → (𝑓 ⊆ 𝒫 𝑗𝐹 ⊆ 𝒫 𝑋))
1512, 14anbi12d 632 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → ((((nei‘𝑗)‘{𝑥}) ⊆ 𝑓𝑓 ⊆ 𝒫 𝑗) ↔ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹𝐹 ⊆ 𝒫 𝑋)))
168, 15rabeqbidv 3455 . . 3 ((𝑗 = 𝐽𝑓 = 𝐹) → {𝑥 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓𝑓 ⊆ 𝒫 𝑗)} = {𝑥𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹𝐹 ⊆ 𝒫 𝑋)})
17 df-flim 23947 . . 3 fLim = (𝑗 ∈ Top, 𝑓 ran Fil ↦ {𝑥 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓𝑓 ⊆ 𝒫 𝑗)})
1816, 17ovmpoga 7587 . 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 1464 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 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  wss 3951  𝒫 cpw 4600  {csn 4626   cuni 4907  ran crn 5686  cfv 6561  (class class class)co 7431  Topctop 22899  neicnei 23105  Filcfil 23853   fLim cflim 23942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-top 22900  df-flim 23947
This theorem is referenced by:  elflim2  23972
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