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Theorem cfilufg 21907
 Description: The filter generated by a Cauchy filter base is still a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.)
Assertion
Ref Expression
cfilufg ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → (𝑋filGen𝐹) ∈ (CauFilu𝑈))

Proof of Theorem cfilufg
Dummy variables 𝑎 𝑏 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfilufbas 21903 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → 𝐹 ∈ (fBas‘𝑋))
2 fgcl 21492 . . 3 (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
3 filfbas 21462 . . 3 ((𝑋filGen𝐹) ∈ (Fil‘𝑋) → (𝑋filGen𝐹) ∈ (fBas‘𝑋))
41, 2, 33syl 18 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → (𝑋filGen𝐹) ∈ (fBas‘𝑋))
51ad3antrrr 762 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) ∧ 𝑏𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝐹 ∈ (fBas‘𝑋))
6 ssfg 21486 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
75, 6syl 17 . . . . . 6 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) ∧ 𝑏𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝐹 ⊆ (𝑋filGen𝐹))
8 simplr 788 . . . . . 6 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) ∧ 𝑏𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝑏𝐹)
97, 8sseldd 3569 . . . . 5 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) ∧ 𝑏𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝑏 ∈ (𝑋filGen𝐹))
10 id 22 . . . . . . . 8 (𝑎 = 𝑏𝑎 = 𝑏)
1110sqxpeqd 5065 . . . . . . 7 (𝑎 = 𝑏 → (𝑎 × 𝑎) = (𝑏 × 𝑏))
1211sseq1d 3595 . . . . . 6 (𝑎 = 𝑏 → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ (𝑏 × 𝑏) ⊆ 𝑣))
1312rspcev 3282 . . . . 5 ((𝑏 ∈ (𝑋filGen𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
149, 13sylancom 698 . . . 4 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) ∧ 𝑏𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
15 iscfilu 21902 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑏𝐹 (𝑏 × 𝑏) ⊆ 𝑣)))
1615simplbda 652 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → ∀𝑣𝑈𝑏𝐹 (𝑏 × 𝑏) ⊆ 𝑣)
1716r19.21bi 2916 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) → ∃𝑏𝐹 (𝑏 × 𝑏) ⊆ 𝑣)
1814, 17r19.29a 3060 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) → ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
1918ralrimiva 2949 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → ∀𝑣𝑈𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
20 iscfilu 21902 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → ((𝑋filGen𝐹) ∈ (CauFilu𝑈) ↔ ((𝑋filGen𝐹) ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)))
2120adantr 480 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → ((𝑋filGen𝐹) ∈ (CauFilu𝑈) ↔ ((𝑋filGen𝐹) ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)))
224, 19, 21mpbir2and 959 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → (𝑋filGen𝐹) ∈ (CauFilu𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   × cxp 5036  ‘cfv 5804  (class class class)co 6549  fBascfbas 19555  filGencfg 19556  Filcfil 21459  UnifOncust 21813  CauFiluccfilu 21900 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-fbas 19564  df-fg 19565  df-fil 21460  df-ust 21814  df-cfilu 21901 This theorem is referenced by:  ucnextcn  21918
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