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Theorem iscfilu 22002
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 21938 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
2 unieq 4410 . . . . . . . . 9 (𝑢 = 𝑈 𝑢 = 𝑈)
32dmeqd 5286 . . . . . . . 8 (𝑢 = 𝑈 → dom 𝑢 = dom 𝑈)
43fveq2d 6152 . . . . . . 7 (𝑢 = 𝑈 → (fBas‘dom 𝑢) = (fBas‘dom 𝑈))
5 raleq 3127 . . . . . . 7 (𝑢 = 𝑈 → (∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣))
64, 5rabeqbidv 3181 . . . . . 6 (𝑢 = 𝑈 → {𝑓 ∈ (fBas‘dom 𝑢) ∣ ∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣} = {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
7 df-cfilu 22001 . . . . . 6 CauFilu = (𝑢 ran UnifOn ↦ {𝑓 ∈ (fBas‘dom 𝑢) ∣ ∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
8 fvex 6158 . . . . . . 7 (fBas‘dom 𝑈) ∈ V
98rabex 4773 . . . . . 6 {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣} ∈ V
106, 7, 9fvmpt 6239 . . . . 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 2684 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ 𝐹 ∈ {𝑓 ∈ (fBas‘dom 𝑈) ∣ ∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣}))
13 rexeq 3128 . . . . 5 (𝑓 = 𝐹 → (∃𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
1413ralbidv 2980 . . . 4 (𝑓 = 𝐹 → (∀𝑣𝑈𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣))
1514elrab 3346 . . 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 21939 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
1817fveq2d 6152 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (fBas‘𝑋) = (fBas‘dom 𝑈))
1918eleq2d 2684 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (fBas‘𝑋) ↔ 𝐹 ∈ (fBas‘dom 𝑈)))
2019anbi1d 740 . 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 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  wss 3555   cuni 4402   × cxp 5072  dom cdm 5074  ran crn 5075  cfv 5847  fBascfbas 19653  UnifOncust 21913  CauFiluccfilu 22000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-ust 21914  df-cfilu 22001
This theorem is referenced by:  cfilufbas  22003  cfiluexsm  22004  fmucnd  22006  cfilufg  22007  trcfilu  22008  cfiluweak  22009  neipcfilu  22010  cfilucfil  22274
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