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Theorem cfiluexsm 22617
Description: For a Cauchy filter base and any entourage 𝑉, there is an element of the filter small in 𝑉. (Contributed by Thierry Arnoux, 19-Nov-2017.)
Assertion
Ref Expression
cfiluexsm ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈) ∧ 𝑉𝑈) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑉)
Distinct variable groups:   𝐹,𝑎   𝑉,𝑎
Allowed substitution hints:   𝑈(𝑎)   𝑋(𝑎)

Proof of Theorem cfiluexsm
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 iscfilu 22615 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
21simplbda 492 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
323adant3 1113 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈) ∧ 𝑉𝑈) → ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
4 sseq2 3876 . . . . 5 (𝑣 = 𝑉 → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ (𝑎 × 𝑎) ⊆ 𝑉))
54rexbidv 3235 . . . 4 (𝑣 = 𝑉 → (∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑉))
65rspcv 3524 . . 3 (𝑉𝑈 → (∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣 → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑉))
763ad2ant3 1116 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈) ∧ 𝑉𝑈) → (∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣 → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑉))
83, 7mpd 15 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈) ∧ 𝑉𝑈) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1069   = wceq 1508  wcel 2051  wral 3081  wrex 3082  wss 3822   × cxp 5401  cfv 6185  fBascfbas 20250  UnifOncust 22526  CauFiluccfilu 22613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-iota 6149  df-fun 6187  df-fn 6188  df-fv 6193  df-ust 22527  df-cfilu 22614
This theorem is referenced by:  fmucnd  22619  cfilucfil  22887
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