MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscfilu Structured version   Visualization version   GIF version

Theorem iscfilu 24313
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 6939 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
2 unieq 4923 . . . . . . . . 9 (𝑢 = 𝑈 𝑢 = 𝑈)
32dmeqd 5919 . . . . . . . 8 (𝑢 = 𝑈 → dom 𝑢 = dom 𝑈)
43fveq2d 6911 . . . . . . 7 (𝑢 = 𝑈 → (fBas‘dom 𝑢) = (fBas‘dom 𝑈))
5 raleq 3321 . . . . . . 7 (𝑢 = 𝑈 → (∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣))
64, 5rabeqbidv 3452 . . . . . 6 (𝑢 = 𝑈 → {𝑓 ∈ (fBas‘dom 𝑢) ∣ ∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣} = {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
7 df-cfilu 24312 . . . . . 6 CauFilu = (𝑢 ran UnifOn ↦ {𝑓 ∈ (fBas‘dom 𝑢) ∣ ∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
8 fvex 6920 . . . . . . 7 (fBas‘dom 𝑈) ∈ V
98rabex 5345 . . . . . 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 2825 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ 𝐹 ∈ {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣}))
13 rexeq 3320 . . . . 5 (𝑓 = 𝐹 → (∃𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
1413ralbidv 3176 . . . 4 (𝑓 = 𝐹 → (∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
1514elrab 3695 . . 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 24250 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
1817fveq2d 6911 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (fBas‘𝑋) = (fBas‘dom 𝑈))
1918eleq2d 2825 . . 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 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  wss 3963   cuni 4912   × cxp 5687  dom cdm 5689  ran crn 5690  cfv 6563  fBascfbas 21370  UnifOncust 24224  CauFiluccfilu 24311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-ust 24225  df-cfilu 24312
This theorem is referenced by:  cfilufbas  24314  cfiluexsm  24315  fmucnd  24317  cfilufg  24318  trcfilu  24319  cfiluweak  24320  neipcfilu  24321  cfilucfil  24588
  Copyright terms: Public domain W3C validator