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Theorem cfiluexsm 22896
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 22894 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
21simplbda 503 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
323adant3 1129 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈) ∧ 𝑉𝑈) → ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
4 sseq2 3941 . . . . 5 (𝑣 = 𝑉 → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ (𝑎 × 𝑎) ⊆ 𝑉))
54rexbidv 3256 . . . 4 (𝑣 = 𝑉 → (∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑉))
65rspcv 3566 . . 3 (𝑉𝑈 → (∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣 → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑉))
763ad2ant3 1132 . 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 1084   = wceq 1538  wcel 2111  wral 3106  wrex 3107  wss 3881   × cxp 5517  cfv 6324  fBascfbas 20079  UnifOncust 22805  CauFiluccfilu 22892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332  df-ust 22806  df-cfilu 22893
This theorem is referenced by:  fmucnd  22898  cfilucfil  23166
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