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Theorem iscfilu 23592
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 6871 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
2 unieq 4874 . . . . . . . . 9 (𝑢 = 𝑈 𝑢 = 𝑈)
32dmeqd 5859 . . . . . . . 8 (𝑢 = 𝑈 → dom 𝑢 = dom 𝑈)
43fveq2d 6843 . . . . . . 7 (𝑢 = 𝑈 → (fBas‘dom 𝑢) = (fBas‘dom 𝑈))
5 raleq 3307 . . . . . . 7 (𝑢 = 𝑈 → (∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣))
64, 5rabeqbidv 3422 . . . . . 6 (𝑢 = 𝑈 → {𝑓 ∈ (fBas‘dom 𝑢) ∣ ∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣} = {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
7 df-cfilu 23591 . . . . . 6 CauFilu = (𝑢 ran UnifOn ↦ {𝑓 ∈ (fBas‘dom 𝑢) ∣ ∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
8 fvex 6852 . . . . . . 7 (fBas‘dom 𝑈) ∈ V
98rabex 5287 . . . . . 6 {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣} ∈ V
106, 7, 9fvmpt 6945 . . . . 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 2823 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ 𝐹 ∈ {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣}))
13 rexeq 3308 . . . . 5 (𝑓 = 𝐹 → (∃𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
1413ralbidv 3172 . . . 4 (𝑓 = 𝐹 → (∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
1514elrab 3643 . . 3 (𝐹 ∈ {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣} ↔ (𝐹 ∈ (fBas‘dom 𝑈) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
1612, 15bitrdi 286 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘dom 𝑈) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
17 ustbas2 23529 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
1817fveq2d 6843 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (fBas‘𝑋) = (fBas‘dom 𝑈))
1918eleq2d 2823 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (fBas‘𝑋) ↔ 𝐹 ∈ (fBas‘dom 𝑈)))
2019anbi1d 630 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ((𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣) ↔ (𝐹 ∈ (fBas‘dom 𝑈) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
2116, 20bitr4d 281 1 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3062  wrex 3071  {crab 3405  wss 3908   cuni 4863   × cxp 5629  dom cdm 5631  ran crn 5632  cfv 6493  fBascfbas 20737  UnifOncust 23503  CauFiluccfilu 23590
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 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  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-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6445  df-fun 6495  df-fv 6501  df-ust 23504  df-cfilu 23591
This theorem is referenced by:  cfilufbas  23593  cfiluexsm  23594  fmucnd  23596  cfilufg  23597  trcfilu  23598  cfiluweak  23599  neipcfilu  23600  cfilucfil  23867
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