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Theorem cfiluweak 24320
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 24254 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
2 iscfilu 24313 . . . . . 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 1773 . . . 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 23856 . . . . 5 (𝐹 ∈ (fBas‘𝐴) → 𝐹 ⊆ 𝒫 𝐴)
75, 6syl 17 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ⊆ 𝒫 𝐴)
8 simp2 1136 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐴𝑋)
98sspwd 4618 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
107, 9sstrd 4006 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ⊆ 𝒫 𝑋)
11 simp1 1135 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝑈 ∈ (UnifOn‘𝑋))
1211elfvexd 6946 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝑋 ∈ V)
13 fbasweak 23889 . . 3 ((𝐹 ∈ (fBas‘𝐴) ∧ 𝐹 ⊆ 𝒫 𝑋𝑋 ∈ V) → 𝐹 ∈ (fBas‘𝑋))
145, 10, 12, 13syl3anc 1370 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ∈ (fBas‘𝑋))
15 sseq2 4022 . . . . . 6 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑎 × 𝑎) ⊆ 𝑢 ↔ (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴))))
1615rexbidv 3177 . . . . 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 5330 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝐴 ∈ V)
2322, 22xpexd 7770 . . . . . 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 17475 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V ∧ 𝑣𝑈) → (𝑣 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
2619, 23, 24, 25syl3anc 1370 . . . . 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 4245 . . . . . 6 (𝑣 ∩ (𝐴 × 𝐴)) ⊆ 𝑣
29 sstr 4004 . . . . . 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 3082 . . . 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 3144 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
34 iscfilu 24313 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
35343ad2ant1 1132 . 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 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  cin 3962  wss 3963  𝒫 cpw 4605   × cxp 5687  cfv 6563  (class class class)co 7431  t crest 17467  fBascfbas 21370  UnifOncust 24224  CauFiluccfilu 24311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-rest 17469  df-fbas 21379  df-ust 24225  df-cfilu 24312
This theorem is referenced by: (None)
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