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Theorem cfilufg 22505
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 22501 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → 𝐹 ∈ (fBas‘𝑋))
2 fgcl 22090 . . 3 (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
3 filfbas 22060 . . 3 ((𝑋filGen𝐹) ∈ (Fil‘𝑋) → (𝑋filGen𝐹) ∈ (fBas‘𝑋))
41, 2, 33syl 18 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → (𝑋filGen𝐹) ∈ (fBas‘𝑋))
51ad3antrrr 720 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) ∧ 𝑏𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝐹 ∈ (fBas‘𝑋))
6 ssfg 22084 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
75, 6syl 17 . . . . . 6 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) ∧ 𝑏𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝐹 ⊆ (𝑋filGen𝐹))
8 simplr 759 . . . . . 6 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) ∧ 𝑏𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝑏𝐹)
97, 8sseldd 3822 . . . . 5 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) ∧ 𝑏𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → 𝑏 ∈ (𝑋filGen𝐹))
10 id 22 . . . . . . . 8 (𝑎 = 𝑏𝑎 = 𝑏)
1110sqxpeqd 5387 . . . . . . 7 (𝑎 = 𝑏 → (𝑎 × 𝑎) = (𝑏 × 𝑏))
1211sseq1d 3851 . . . . . 6 (𝑎 = 𝑏 → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ (𝑏 × 𝑏) ⊆ 𝑣))
1312rspcev 3511 . . . . 5 ((𝑏 ∈ (𝑋filGen𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
149, 13sylancom 582 . . . 4 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) ∧ 𝑏𝐹) ∧ (𝑏 × 𝑏) ⊆ 𝑣) → ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
15 iscfilu 22500 . . . . . 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 3114 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) → ∃𝑏𝐹 (𝑏 × 𝑏) ⊆ 𝑣)
1814, 17r19.29a 3264 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) ∧ 𝑣𝑈) → ∃𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
1918ralrimiva 3148 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → ∀𝑣𝑈𝑎 ∈ (𝑋filGen𝐹)(𝑎 × 𝑎) ⊆ 𝑣)
20 iscfilu 22500 . . 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 703 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → (𝑋filGen𝐹) ∈ (CauFilu𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wcel 2107  wral 3090  wrex 3091  wss 3792   × cxp 5353  cfv 6135  (class class class)co 6922  fBascfbas 20130  filGencfg 20131  Filcfil 22057  UnifOncust 22411  CauFiluccfilu 22498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-fbas 20139  df-fg 20140  df-fil 22058  df-ust 22412  df-cfilu 22499
This theorem is referenced by:  ucnextcn  22516
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