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Theorem iscfilu 24297
Description: The predicate "𝐹 is a Cauchy filter base on uniform space 𝑈". (Contributed by Thierry Arnoux, 18-Nov-2017.)
Assertion
Ref Expression
iscfilu (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
Distinct variable groups:   𝑣,𝑎,𝐹   𝑣,𝑈
Allowed substitution hints:   𝑈(𝑎)   𝑋(𝑣,𝑎)

Proof of Theorem iscfilu
Dummy variables 𝑓 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvunirn 6938 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
2 unieq 4918 . . . . . . . . 9 (𝑢 = 𝑈 𝑢 = 𝑈)
32dmeqd 5916 . . . . . . . 8 (𝑢 = 𝑈 → dom 𝑢 = dom 𝑈)
43fveq2d 6910 . . . . . . 7 (𝑢 = 𝑈 → (fBas‘dom 𝑢) = (fBas‘dom 𝑈))
5 raleq 3323 . . . . . . 7 (𝑢 = 𝑈 → (∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣))
64, 5rabeqbidv 3455 . . . . . 6 (𝑢 = 𝑈 → {𝑓 ∈ (fBas‘dom 𝑢) ∣ ∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣} = {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
7 df-cfilu 24296 . . . . . 6 CauFilu = (𝑢 ran UnifOn ↦ {𝑓 ∈ (fBas‘dom 𝑢) ∣ ∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
8 fvex 6919 . . . . . . 7 (fBas‘dom 𝑈) ∈ V
98rabex 5339 . . . . . 6 {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣} ∈ V
106, 7, 9fvmpt 7016 . . . . 5 (𝑈 ran UnifOn → (CauFilu𝑈) = {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
111, 10syl 17 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (CauFilu𝑈) = {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
1211eleq2d 2827 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ 𝐹 ∈ {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣}))
13 rexeq 3322 . . . . 5 (𝑓 = 𝐹 → (∃𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
1413ralbidv 3178 . . . 4 (𝑓 = 𝐹 → (∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
1514elrab 3692 . . 3 (𝐹 ∈ {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣} ↔ (𝐹 ∈ (fBas‘dom 𝑈) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
1612, 15bitrdi 287 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘dom 𝑈) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
17 ustbas2 24234 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
1817fveq2d 6910 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (fBas‘𝑋) = (fBas‘dom 𝑈))
1918eleq2d 2827 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (fBas‘𝑋) ↔ 𝐹 ∈ (fBas‘dom 𝑈)))
2019anbi1d 631 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ((𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣) ↔ (𝐹 ∈ (fBas‘dom 𝑈) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
2116, 20bitr4d 282 1 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  wss 3951   cuni 4907   × cxp 5683  dom cdm 5685  ran crn 5686  cfv 6561  fBascfbas 21352  UnifOncust 24208  CauFiluccfilu 24295
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-pow 5365  ax-pr 5432  ax-un 7755
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-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  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-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ust 24209  df-cfilu 24296
This theorem is referenced by:  cfilufbas  24298  cfiluexsm  24299  fmucnd  24301  cfilufg  24302  trcfilu  24303  cfiluweak  24304  neipcfilu  24305  cfilucfil  24572
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