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Theorem iscusp2 22326
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 22323 . 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 6336 . . . 4 (Fil‘𝐵) = (Fil‘(Base‘𝑊))
4 iscusp2.2 . . . . . . 7 𝑈 = (UnifSt‘𝑊)
54fveq2i 6336 . . . . . 6 (CauFilu𝑈) = (CauFilu‘(UnifSt‘𝑊))
65eleq2i 2842 . . . . 5 (𝑐 ∈ (CauFilu𝑈) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)))
7 iscusp2.3 . . . . . . 7 𝐽 = (TopOpen‘𝑊)
87oveq1i 6806 . . . . . 6 (𝐽 fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐)
98neeq1i 3007 . . . . 5 ((𝐽 fLim 𝑐) ≠ ∅ ↔ ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)
106, 9imbi12i 339 . . . 4 ((𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
113, 10raleqbii 3139 . . 3 (∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
1211anbi2i 609 . 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 267 1 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  c0 4063  cfv 6030  (class class class)co 6796  Basecbs 16064  TopOpenctopn 16290  Filcfil 21869   fLim cflim 21958  UnifStcuss 22277  UnifSpcusp 22278  CauFiluccfilu 22310  CUnifSpccusp 22321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038  df-ov 6799  df-cusp 22322
This theorem is referenced by:  cmetcusp1  23368
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