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Theorem cuspcvg 22475
Description: In a complete uniform space, any Cauchy filter 𝐶 has a limit. (Contributed by Thierry Arnoux, 3-Dec-2017.)
Hypotheses
Ref Expression
cuspcvg.1 𝐵 = (Base‘𝑊)
cuspcvg.2 𝐽 = (TopOpen‘𝑊)
Assertion
Ref Expression
cuspcvg ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐽 fLim 𝐶) ≠ ∅)

Proof of Theorem cuspcvg
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2894 . . . . 5 (𝑐 = 𝐶 → (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) ↔ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))))
2 cuspcvg.2 . . . . . . . . 9 𝐽 = (TopOpen‘𝑊)
32eqcomi 2834 . . . . . . . 8 (TopOpen‘𝑊) = 𝐽
43a1i 11 . . . . . . 7 (𝑐 = 𝐶 → (TopOpen‘𝑊) = 𝐽)
5 id 22 . . . . . . 7 (𝑐 = 𝐶𝑐 = 𝐶)
64, 5oveq12d 6923 . . . . . 6 (𝑐 = 𝐶 → ((TopOpen‘𝑊) fLim 𝑐) = (𝐽 fLim 𝐶))
76neeq1d 3058 . . . . 5 (𝑐 = 𝐶 → (((TopOpen‘𝑊) fLim 𝑐) ≠ ∅ ↔ (𝐽 fLim 𝐶) ≠ ∅))
81, 7imbi12d 336 . . . 4 (𝑐 = 𝐶 → ((𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅) ↔ (𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) → (𝐽 fLim 𝐶) ≠ ∅)))
9 iscusp 22473 . . . . . 6 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
109simprbi 492 . . . . 5 (𝑊 ∈ CUnifSp → ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
1110adantr 474 . . . 4 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
12 simpr 479 . . . . 5 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘𝐵))
13 cuspcvg.1 . . . . . 6 𝐵 = (Base‘𝑊)
1413fveq2i 6436 . . . . 5 (Fil‘𝐵) = (Fil‘(Base‘𝑊))
1512, 14syl6eleq 2916 . . . 4 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘(Base‘𝑊)))
168, 11, 15rspcdva 3532 . . 3 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) → (𝐽 fLim 𝐶) ≠ ∅))
17163impia 1151 . 2 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵) ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))) → (𝐽 fLim 𝐶) ≠ ∅)
18173com23 1162 1 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐽 fLim 𝐶) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113   = wceq 1658  wcel 2166  wne 2999  wral 3117  c0 4144  cfv 6123  (class class class)co 6905  Basecbs 16222  TopOpenctopn 16435  Filcfil 22019   fLim cflim 22108  UnifStcuss 22427  UnifSpcusp 22428  CauFiluccfilu 22460  CUnifSpccusp 22471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-cusp 22472
This theorem is referenced by:  cnextucn  22477
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