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Theorem cfilufg 22467
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 22463 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → 𝐹 ∈ (fBas‘𝑋))
2 fgcl 22052 . . 3 (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
3 filfbas 22022 . . 3 ((𝑋filGen𝐹) ∈ (Fil‘𝑋) → (𝑋filGen𝐹) ∈ (fBas‘𝑋))
41, 2, 33syl 18 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → (𝑋filGen𝐹) ∈ (fBas‘𝑋))
51ad3antrrr 723 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) ∧ 𝑏𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝐹 ∈ (fBas‘𝑋))
6 ssfg 22046 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
75, 6syl 17 . . . . . 6 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) ∧ 𝑏𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝐹 ⊆ (𝑋filGen𝐹))
8 simplr 787 . . . . . 6 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) ∧ 𝑏𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝑏𝐹)
97, 8sseldd 3828 . . . . 5 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) ∧ 𝑏𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝑏 ∈ (𝑋filGen𝐹))
10 id 22 . . . . . . . 8 (𝑎 = 𝑏𝑎 = 𝑏)
1110sqxpeqd 5374 . . . . . . 7 (𝑎 = 𝑏 → (𝑎 × 𝑎) = (𝑏 × 𝑏))
1211sseq1d 3857 . . . . . 6 (𝑎 = 𝑏 → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ (𝑏 × 𝑏) ⊆ 𝑣))
1312rspcev 3526 . . . . 5 ((𝑏 ∈ (𝑋filGen𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
149, 13sylancom 584 . . . 4 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) ∧ 𝑏𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
15 iscfilu 22462 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑏𝐹 (𝑏 × 𝑏) ⊆ 𝑣)))
1615simplbda 495 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → ∀𝑣𝑈𝑏𝐹 (𝑏 × 𝑏) ⊆ 𝑣)
1716r19.21bi 3141 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) → ∃𝑏𝐹 (𝑏 × 𝑏) ⊆ 𝑣)
1814, 17r19.29a 3288 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) → ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
1918ralrimiva 3175 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → ∀𝑣𝑈𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
20 iscfilu 22462 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → ((𝑋filGen𝐹) ∈ (CauFilu𝑈) ↔ ((𝑋filGen𝐹) ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)))
2120adantr 474 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → ((𝑋filGen𝐹) ∈ (CauFilu𝑈) ↔ ((𝑋filGen𝐹) ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)))
224, 19, 21mpbir2and 706 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → (𝑋filGen𝐹) ∈ (CauFilu𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wcel 2166  wral 3117  wrex 3118  wss 3798   × cxp 5340  cfv 6123  (class class class)co 6905  fBascfbas 20094  filGencfg 20095  Filcfil 22019  UnifOncust 22373  CauFiluccfilu 22460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-fbas 20103  df-fg 20104  df-fil 22020  df-ust 22374  df-cfilu 22461
This theorem is referenced by:  ucnextcn  22478
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