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Theorem flimval 21987
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 20931 . . . 4 (𝐽 ∈ Top → 𝑋𝐽)
32adantr 466 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → 𝑋𝐽)
4 rabexg 4945 . . 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 468 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
76unieqd 4584 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝐽)
87, 1syl6eqr 2823 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑗 = 𝑋)
96fveq2d 6336 . . . . . . 7 ((𝑗 = 𝐽𝑓 = 𝐹) → (nei‘𝑗) = (nei‘𝐽))
109fveq1d 6334 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
11 simpr 471 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑓 = 𝐹)
1210, 11sseq12d 3783 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ↔ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹))
138pweqd 4302 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝒫 𝑗 = 𝒫 𝑋)
1411, 13sseq12d 3783 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → (𝑓 ⊆ 𝒫 𝑗𝐹 ⊆ 𝒫 𝑋))
1512, 14anbi12d 616 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → ((((nei‘𝑗)‘{𝑥}) ⊆ 𝑓𝑓 ⊆ 𝒫 𝑗) ↔ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹𝐹 ⊆ 𝒫 𝑋)))
168, 15rabeqbidv 3345 . . 3 ((𝑗 = 𝐽𝑓 = 𝐹) → {𝑥 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓𝑓 ⊆ 𝒫 𝑗)} = {𝑥𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹𝐹 ⊆ 𝒫 𝑋)})
17 df-flim 21963 . . 3 fLim = (𝑗 ∈ Top, 𝑓 ran Fil ↦ {𝑥 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓𝑓 ⊆ 𝒫 𝑗)})
1816, 17ovmpt2ga 6937 . 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 1573 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 382   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351  wss 3723  𝒫 cpw 4297  {csn 4316   cuni 4574  ran crn 5250  cfv 6031  (class class class)co 6793  Topctop 20918  neicnei 21122  Filcfil 21869   fLim cflim 21958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-top 20919  df-flim 21963
This theorem is referenced by:  elflim2  21988
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