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Theorem cfiluweak 22320
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 22254 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
2 iscfilu 22313 . . . . . 6 ((𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → (𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴))) ↔ (𝐹 ∈ (fBas‘𝐴) ∧ ∀𝑢 ∈ (𝑈t (𝐴 × 𝐴))∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢)))
32biimpa 502 . . . . 5 (((𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) ∧ 𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → (𝐹 ∈ (fBas‘𝐴) ∧ ∀𝑢 ∈ (𝑈t (𝐴 × 𝐴))∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢))
41, 3stoic3 1850 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → (𝐹 ∈ (fBas‘𝐴) ∧ ∀𝑢 ∈ (𝑈t (𝐴 × 𝐴))∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢))
54simpld 477 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ∈ (fBas‘𝐴))
6 fbsspw 21857 . . . . 5 (𝐹 ∈ (fBas‘𝐴) → 𝐹 ⊆ 𝒫 𝐴)
75, 6syl 17 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ⊆ 𝒫 𝐴)
8 simp2 1132 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐴𝑋)
9 sspwb 5066 . . . . 5 (𝐴𝑋 ↔ 𝒫 𝐴 ⊆ 𝒫 𝑋)
108, 9sylib 208 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
117, 10sstrd 3754 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ⊆ 𝒫 𝑋)
12 simp1 1131 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝑈 ∈ (UnifOn‘𝑋))
1312elfvexd 6384 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝑋 ∈ V)
14 fbasweak 21890 . . 3 ((𝐹 ∈ (fBas‘𝐴) ∧ 𝐹 ⊆ 𝒫 𝑋𝑋 ∈ V) → 𝐹 ∈ (fBas‘𝑋))
155, 11, 13, 14syl3anc 1477 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ∈ (fBas‘𝑋))
16 sseq2 3768 . . . . . 6 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → ((𝑎 × 𝑎) ⊆ 𝑢 ↔ (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴))))
1716rexbidv 3190 . . . . 5 (𝑢 = (𝑣 ∩ (𝐴 × 𝐴)) → (∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢 ↔ ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴))))
184simprd 482 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → ∀𝑢 ∈ (𝑈t (𝐴 × 𝐴))∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢)
1918adantr 472 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → ∀𝑢 ∈ (𝑈t (𝐴 × 𝐴))∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑢)
2012adantr 472 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
2113adantr 472 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝑋 ∈ V)
228adantr 472 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝐴𝑋)
2321, 22ssexd 4957 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝐴 ∈ V)
24 xpexg 7126 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐴 ∈ V) → (𝐴 × 𝐴) ∈ V)
2523, 23, 24syl2anc 696 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → (𝐴 × 𝐴) ∈ V)
26 simpr 479 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → 𝑣𝑈)
27 elrestr 16311 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V ∧ 𝑣𝑈) → (𝑣 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
2820, 25, 26, 27syl3anc 1477 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → (𝑣 ∩ (𝐴 × 𝐴)) ∈ (𝑈t (𝐴 × 𝐴)))
2917, 19, 28rspcdva 3455 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)))
30 inss1 3976 . . . . . 6 (𝑣 ∩ (𝐴 × 𝐴)) ⊆ 𝑣
31 sstr 3752 . . . . . 6 (((𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)) ∧ (𝑣 ∩ (𝐴 × 𝐴)) ⊆ 𝑣) → (𝑎 × 𝑎) ⊆ 𝑣)
3230, 31mpan2 709 . . . . 5 ((𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)) → (𝑎 × 𝑎) ⊆ 𝑣)
3332reximi 3149 . . . 4 (∃𝑎𝐹 (𝑎 × 𝑎) ⊆ (𝑣 ∩ (𝐴 × 𝐴)) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3429, 33syl 17 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) ∧ 𝑣𝑈) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
3534ralrimiva 3104 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
36 iscfilu 22313 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
37363ad2ant1 1128 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
3815, 35, 37mpbir2and 995 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ∈ (CauFilu𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  cin 3714  wss 3715  𝒫 cpw 4302   × cxp 5264  cfv 6049  (class class class)co 6814  t crest 16303  fBascfbas 19956  UnifOncust 22224  CauFiluccfilu 22311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-rest 16305  df-fbas 19965  df-ust 22225  df-cfilu 22312
This theorem is referenced by: (None)
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