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Theorem iscusp 22304
 Description: The predicate "𝑊 is a complete uniform space." (Contributed by Thierry Arnoux, 3-Dec-2017.)
Assertion
Ref Expression
iscusp (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
Distinct variable group:   𝑊,𝑐

Proof of Theorem iscusp
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6352 . . . 4 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
21fveq2d 6356 . . 3 (𝑤 = 𝑊 → (Fil‘(Base‘𝑤)) = (Fil‘(Base‘𝑊)))
3 fveq2 6352 . . . . . 6 (𝑤 = 𝑊 → (UnifSt‘𝑤) = (UnifSt‘𝑊))
43fveq2d 6356 . . . . 5 (𝑤 = 𝑊 → (CauFilu‘(UnifSt‘𝑤)) = (CauFilu‘(UnifSt‘𝑊)))
54eleq2d 2825 . . . 4 (𝑤 = 𝑊 → (𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊))))
6 fveq2 6352 . . . . . 6 (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
76oveq1d 6828 . . . . 5 (𝑤 = 𝑊 → ((TopOpen‘𝑤) fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐))
87neeq1d 2991 . . . 4 (𝑤 = 𝑊 → (((TopOpen‘𝑤) fLim 𝑐) ≠ ∅ ↔ ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
95, 8imbi12d 333 . . 3 (𝑤 = 𝑊 → ((𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅) ↔ (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
102, 9raleqbidv 3291 . 2 (𝑤 = 𝑊 → (∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅) ↔ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
11 df-cusp 22303 . 2 CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)}
1210, 11elrab2 3507 1 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ≠ wne 2932  ∀wral 3050  ∅c0 4058  ‘cfv 6049  (class class class)co 6813  Basecbs 16059  TopOpenctopn 16284  Filcfil 21850   fLim cflim 21939  UnifStcuss 22258  UnifSpcusp 22259  CauFiluccfilu 22291  CUnifSpccusp 22302 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6816  df-cusp 22303 This theorem is referenced by:  cuspusp  22305  cuspcvg  22306  iscusp2  22307  cmetcusp  23350
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