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Theorem iscusp2 22518
Description: The predicate "𝑊 is a complete uniform space." (Contributed by Thierry Arnoux, 15-Dec-2017.)
Hypotheses
Ref Expression
iscusp2.1 𝐵 = (Base‘𝑊)
iscusp2.2 𝑈 = (UnifSt‘𝑊)
iscusp2.3 𝐽 = (TopOpen‘𝑊)
Assertion
Ref Expression
iscusp2 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅)))
Distinct variable group:   𝑊,𝑐
Allowed substitution hints:   𝐵(𝑐)   𝑈(𝑐)   𝐽(𝑐)

Proof of Theorem iscusp2
StepHypRef Expression
1 iscusp 22515 . 2 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
2 iscusp2.1 . . . . 5 𝐵 = (Base‘𝑊)
32fveq2i 6451 . . . 4 (Fil‘𝐵) = (Fil‘(Base‘𝑊))
4 iscusp2.2 . . . . . . 7 𝑈 = (UnifSt‘𝑊)
54fveq2i 6451 . . . . . 6 (CauFilu𝑈) = (CauFilu‘(UnifSt‘𝑊))
65eleq2i 2851 . . . . 5 (𝑐 ∈ (CauFilu𝑈) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)))
7 iscusp2.3 . . . . . . 7 𝐽 = (TopOpen‘𝑊)
87oveq1i 6934 . . . . . 6 (𝐽 fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐)
98neeq1i 3033 . . . . 5 ((𝐽 fLim 𝑐) ≠ ∅ ↔ ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)
106, 9imbi12i 342 . . . 4 ((𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
113, 10raleqbii 3172 . . 3 (∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
1211anbi2i 616 . 2 ((𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅)) ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
131, 12bitr4i 270 1 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  wne 2969  wral 3090  c0 4141  cfv 6137  (class class class)co 6924  Basecbs 16259  TopOpenctopn 16472  Filcfil 22061   fLim cflim 22150  UnifStcuss 22469  UnifSpcusp 22470  CauFiluccfilu 22502  CUnifSpccusp 22513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-iota 6101  df-fv 6145  df-ov 6927  df-cusp 22514
This theorem is referenced by:  cmetcusp1  23563
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