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Theorem cfiluweak 22469
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 22403 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
2 iscfilu 22462 . . . . . 6 ((𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → (𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴))) ↔ (𝐹 ∈ (fBas‘𝐴) ∧ ∀𝑢 ∈ (𝑈t (𝐴 × 𝐴))∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢)))
32biimpa 470 . . . . 5 (((𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) ∧ 𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → (𝐹 ∈ (fBas‘𝐴) ∧ ∀𝑢 ∈ (𝑈t (𝐴 × 𝐴))∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢))
41, 3stoic3 1877 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → (𝐹 ∈ (fBas‘𝐴) ∧ ∀𝑢 ∈ (𝑈t (𝐴 × 𝐴))∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢))
54simpld 490 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ∈ (fBas‘𝐴))
6 fbsspw 22006 . . . . 5 (𝐹 ∈ (fBas‘𝐴) → 𝐹 ⊆ 𝒫 𝐴)
75, 6syl 17 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ⊆ 𝒫 𝐴)
8 simp2 1173 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐴𝑋)
9 sspwb 5138 . . . . 5 (𝐴𝑋 ↔ 𝒫 𝐴 ⊆ 𝒫 𝑋)
108, 9sylib 210 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
117, 10sstrd 3837 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ⊆ 𝒫 𝑋)
12 simp1 1172 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝑈 ∈ (UnifOn‘𝑋))
1312elfvexd 6468 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝑋 ∈ V)
14 fbasweak 22039 . . 3 ((𝐹 ∈ (fBas‘𝐴) ∧ 𝐹 ⊆ 𝒫 𝑋𝑋 ∈ V) → 𝐹 ∈ (fBas‘𝑋))
155, 11, 13, 14syl3anc 1496 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ∈ (fBas‘𝑋))
16 sseq2 3852 . . . . . 6 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑎 × 𝑎) ⊆ 𝑢 ↔ (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴))))
1716rexbidv 3262 . . . . 5 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → (∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢 ↔ ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴))))
184simprd 491 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → ∀𝑢 ∈ (𝑈t (𝐴 × 𝐴))∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢)
1918adantr 474 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → ∀𝑢 ∈ (𝑈t (𝐴 × 𝐴))∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢)
2012adantr 474 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
2113adantr 474 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝑋 ∈ V)
228adantr 474 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝐴𝑋)
2321, 22ssexd 5030 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝐴 ∈ V)
2423, 23xpexd 7221 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → (𝐴 × 𝐴) ∈ V)
25 simpr 479 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝑣𝑈)
26 elrestr 16442 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V ∧ 𝑣𝑈) → (𝑣 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
2720, 24, 25, 26syl3anc 1496 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → (𝑣 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
2817, 19, 27rspcdva 3532 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
29 inss1 4057 . . . . . 6 (𝑣 ∩ (𝐴 × 𝐴)) ⊆ 𝑣
30 sstr 3835 . . . . . 6 (((𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)) ∧ (𝑣 ∩ (𝐴 × 𝐴)) ⊆ 𝑣) → (𝑎 × 𝑎) ⊆ 𝑣)
3129, 30mpan2 684 . . . . 5 ((𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)) → (𝑎 × 𝑎) ⊆ 𝑣)
3231reximi 3219 . . . 4 (∃𝑎𝐹 (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3328, 32syl 17 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3433ralrimiva 3175 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
35 iscfilu 22462 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
36353ad2ant1 1169 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
3715, 34, 36mpbir2and 706 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ∈ (CauFilu𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wral 3117  wrex 3118  Vcvv 3414  cin 3797  wss 3798  𝒫 cpw 4378   × cxp 5340  cfv 6123  (class class class)co 6905  t crest 16434  fBascfbas 20094  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-rep 4994  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-reu 3124  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-iun 4742  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-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-rest 16436  df-fbas 20103  df-ust 22374  df-cfilu 22461
This theorem is referenced by: (None)
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