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Theorem cfiluexsm 24244
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 24242 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
21simplbda 498 . . 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 4003 . . . . 5 (𝑣 = 𝑉 → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ (𝑎 × 𝑎) ⊆ 𝑉))
54rexbidv 3168 . . . 4 (𝑣 = 𝑉 → (∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑉))
65rspcv 3602 . . 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 1533  wcel 2098  wral 3050  wrex 3059  wss 3944   × cxp 5676  cfv 6549  fBascfbas 21289  UnifOncust 24153  CauFiluccfilu 24240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-iota 6501  df-fun 6551  df-fv 6557  df-ust 24154  df-cfilu 24241
This theorem is referenced by:  fmucnd  24246  cfilucfil  24517
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