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Theorem cfiluexsm 22004
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 22002 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
21simplbda 653 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
323adant3 1079 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈) ∧ 𝑉𝑈) → ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)
4 sseq2 3606 . . . . 5 (𝑣 = 𝑉 → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ (𝑎 × 𝑎) ⊆ 𝑉))
54rexbidv 3045 . . . 4 (𝑣 = 𝑉 → (∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑉))
65rspcv 3291 . . 3 (𝑉𝑈 → (∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣 → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑉))
763ad2ant3 1082 . 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 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  wss 3555   × cxp 5072  cfv 5847  fBascfbas 19653  UnifOncust 21913  CauFiluccfilu 22000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-ust 21914  df-cfilu 22001
This theorem is referenced by:  fmucnd  22006  cfilucfil  22274
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