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Theorem cfiluweak 24304
Description: A Cauchy filter base is also a Cauchy filter base on any coarser uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.)
Assertion
Ref Expression
cfiluweak ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ∈ (CauFilu𝑈))

Proof of Theorem cfiluweak
Dummy variables 𝑢 𝑎 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 trust 24238 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
2 iscfilu 24297 . . . . . 6 ((𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → (𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴))) ↔ (𝐹 ∈ (fBas‘𝐴) ∧ ∀𝑢 ∈ (𝑈t (𝐴 × 𝐴))∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢)))
32biimpa 476 . . . . 5 (((𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) ∧ 𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → (𝐹 ∈ (fBas‘𝐴) ∧ ∀𝑢 ∈ (𝑈t (𝐴 × 𝐴))∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢))
41, 3stoic3 1776 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → (𝐹 ∈ (fBas‘𝐴) ∧ ∀𝑢 ∈ (𝑈t (𝐴 × 𝐴))∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢))
54simpld 494 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ∈ (fBas‘𝐴))
6 fbsspw 23840 . . . . 5 (𝐹 ∈ (fBas‘𝐴) → 𝐹 ⊆ 𝒫 𝐴)
75, 6syl 17 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ⊆ 𝒫 𝐴)
8 simp2 1138 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐴𝑋)
98sspwd 4613 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
107, 9sstrd 3994 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ⊆ 𝒫 𝑋)
11 simp1 1137 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝑈 ∈ (UnifOn‘𝑋))
1211elfvexd 6945 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝑋 ∈ V)
13 fbasweak 23873 . . 3 ((𝐹 ∈ (fBas‘𝐴) ∧ 𝐹 ⊆ 𝒫 𝑋𝑋 ∈ V) → 𝐹 ∈ (fBas‘𝑋))
145, 10, 12, 13syl3anc 1373 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ∈ (fBas‘𝑋))
15 sseq2 4010 . . . . . 6 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑎 × 𝑎) ⊆ 𝑢 ↔ (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴))))
1615rexbidv 3179 . . . . 5 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → (∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢 ↔ ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴))))
174simprd 495 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → ∀𝑢 ∈ (𝑈t (𝐴 × 𝐴))∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢)
1817adantr 480 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → ∀𝑢 ∈ (𝑈t (𝐴 × 𝐴))∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢)
1911adantr 480 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
2012adantr 480 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝑋 ∈ V)
218adantr 480 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝐴𝑋)
2220, 21ssexd 5324 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝐴 ∈ V)
2322, 22xpexd 7771 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → (𝐴 × 𝐴) ∈ V)
24 simpr 484 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝑣𝑈)
25 elrestr 17473 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V ∧ 𝑣𝑈) → (𝑣 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
2619, 23, 24, 25syl3anc 1373 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → (𝑣 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
2716, 18, 26rspcdva 3623 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
28 inss1 4237 . . . . . 6 (𝑣 ∩ (𝐴 × 𝐴)) ⊆ 𝑣
29 sstr 3992 . . . . . 6 (((𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)) ∧ (𝑣 ∩ (𝐴 × 𝐴)) ⊆ 𝑣) → (𝑎 × 𝑎) ⊆ 𝑣)
3028, 29mpan2 691 . . . . 5 ((𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)) → (𝑎 × 𝑎) ⊆ 𝑣)
3130reximi 3084 . . . 4 (∃𝑎𝐹 (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3227, 31syl 17 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3332ralrimiva 3146 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
34 iscfilu 24297 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
35343ad2ant1 1134 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
3614, 33, 35mpbir2and 713 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ∈ (CauFilu𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600   × cxp 5683  cfv 6561  (class class class)co 7431  t crest 17465  fBascfbas 21352  UnifOncust 24208  CauFiluccfilu 24295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-rest 17467  df-fbas 21361  df-ust 24209  df-cfilu 24296
This theorem is referenced by: (None)
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