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

Theorem iscfilu 22293
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 elrnust 22229 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
2 unieq 4596 . . . . . . . . 9 (𝑢 = 𝑈 𝑢 = 𝑈)
32dmeqd 5481 . . . . . . . 8 (𝑢 = 𝑈 → dom 𝑢 = dom 𝑈)
43fveq2d 6356 . . . . . . 7 (𝑢 = 𝑈 → (fBas‘dom 𝑢) = (fBas‘dom 𝑈))
5 raleq 3277 . . . . . . 7 (𝑢 = 𝑈 → (∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣))
64, 5rabeqbidv 3335 . . . . . 6 (𝑢 = 𝑈 → {𝑓 ∈ (fBas‘dom 𝑢) ∣ ∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣} = {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
7 df-cfilu 22292 . . . . . 6 CauFilu = (𝑢 ran UnifOn ↦ {𝑓 ∈ (fBas‘dom 𝑢) ∣ ∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
8 fvex 6362 . . . . . . 7 (fBas‘dom 𝑈) ∈ V
98rabex 4964 . . . . . 6 {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣} ∈ V
106, 7, 9fvmpt 6444 . . . . 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 3278 . . . . 5 (𝑓 = 𝐹 → (∃𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
1413ralbidv 3124 . . . 4 (𝑓 = 𝐹 → (∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
1514elrab 3504 . . 3 (𝐹 ∈ {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣} ↔ (𝐹 ∈ (fBas‘dom 𝑈) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
1612, 15syl6bb 276 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘dom 𝑈) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
17 ustbas2 22230 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
1817fveq2d 6356 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (fBas‘𝑋) = (fBas‘dom 𝑈))
1918eleq2d 2825 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (fBas‘𝑋) ↔ 𝐹 ∈ (fBas‘dom 𝑈)))
2019anbi1d 743 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ((𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣) ↔ (𝐹 ∈ (fBas‘dom 𝑈) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
2116, 20bitr4d 271 1 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  wss 3715   cuni 4588   × cxp 5264  dom cdm 5266  ran crn 5267  cfv 6049  fBascfbas 19936  UnifOncust 22204  CauFiluccfilu 22291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-ust 22205  df-cfilu 22292
This theorem is referenced by:  cfilufbas  22294  cfiluexsm  22295  fmucnd  22297  cfilufg  22298  trcfilu  22299  cfiluweak  22300  neipcfilu  22301  cfilucfil  22565
  Copyright terms: Public domain W3C validator