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Theorem iscusp2 23814
Description: The predicate "π‘Š is a complete uniform space." (Contributed by Thierry Arnoux, 15-Dec-2017.)
Hypotheses
Ref Expression
iscusp2.1 𝐡 = (Baseβ€˜π‘Š)
iscusp2.2 π‘ˆ = (UnifStβ€˜π‘Š)
iscusp2.3 𝐽 = (TopOpenβ€˜π‘Š)
Assertion
Ref Expression
iscusp2 (π‘Š ∈ CUnifSp ↔ (π‘Š ∈ UnifSp ∧ βˆ€π‘ ∈ (Filβ€˜π΅)(𝑐 ∈ (CauFiluβ€˜π‘ˆ) β†’ (𝐽 fLim 𝑐) β‰  βˆ…)))
Distinct variable group:   π‘Š,𝑐
Allowed substitution hints:   𝐡(𝑐)   π‘ˆ(𝑐)   𝐽(𝑐)

Proof of Theorem iscusp2
StepHypRef Expression
1 iscusp 23811 . 2 (π‘Š ∈ CUnifSp ↔ (π‘Š ∈ UnifSp ∧ βˆ€π‘ ∈ (Filβ€˜(Baseβ€˜π‘Š))(𝑐 ∈ (CauFiluβ€˜(UnifStβ€˜π‘Š)) β†’ ((TopOpenβ€˜π‘Š) fLim 𝑐) β‰  βˆ…)))
2 iscusp2.1 . . . . 5 𝐡 = (Baseβ€˜π‘Š)
32fveq2i 6894 . . . 4 (Filβ€˜π΅) = (Filβ€˜(Baseβ€˜π‘Š))
4 iscusp2.2 . . . . . . 7 π‘ˆ = (UnifStβ€˜π‘Š)
54fveq2i 6894 . . . . . 6 (CauFiluβ€˜π‘ˆ) = (CauFiluβ€˜(UnifStβ€˜π‘Š))
65eleq2i 2825 . . . . 5 (𝑐 ∈ (CauFiluβ€˜π‘ˆ) ↔ 𝑐 ∈ (CauFiluβ€˜(UnifStβ€˜π‘Š)))
7 iscusp2.3 . . . . . . 7 𝐽 = (TopOpenβ€˜π‘Š)
87oveq1i 7421 . . . . . 6 (𝐽 fLim 𝑐) = ((TopOpenβ€˜π‘Š) fLim 𝑐)
98neeq1i 3005 . . . . 5 ((𝐽 fLim 𝑐) β‰  βˆ… ↔ ((TopOpenβ€˜π‘Š) fLim 𝑐) β‰  βˆ…)
106, 9imbi12i 350 . . . 4 ((𝑐 ∈ (CauFiluβ€˜π‘ˆ) β†’ (𝐽 fLim 𝑐) β‰  βˆ…) ↔ (𝑐 ∈ (CauFiluβ€˜(UnifStβ€˜π‘Š)) β†’ ((TopOpenβ€˜π‘Š) fLim 𝑐) β‰  βˆ…))
113, 10raleqbii 3338 . . 3 (βˆ€π‘ ∈ (Filβ€˜π΅)(𝑐 ∈ (CauFiluβ€˜π‘ˆ) β†’ (𝐽 fLim 𝑐) β‰  βˆ…) ↔ βˆ€π‘ ∈ (Filβ€˜(Baseβ€˜π‘Š))(𝑐 ∈ (CauFiluβ€˜(UnifStβ€˜π‘Š)) β†’ ((TopOpenβ€˜π‘Š) fLim 𝑐) β‰  βˆ…))
1211anbi2i 623 . 2 ((π‘Š ∈ UnifSp ∧ βˆ€π‘ ∈ (Filβ€˜π΅)(𝑐 ∈ (CauFiluβ€˜π‘ˆ) β†’ (𝐽 fLim 𝑐) β‰  βˆ…)) ↔ (π‘Š ∈ UnifSp ∧ βˆ€π‘ ∈ (Filβ€˜(Baseβ€˜π‘Š))(𝑐 ∈ (CauFiluβ€˜(UnifStβ€˜π‘Š)) β†’ ((TopOpenβ€˜π‘Š) fLim 𝑐) β‰  βˆ…)))
131, 12bitr4i 277 1 (π‘Š ∈ CUnifSp ↔ (π‘Š ∈ UnifSp ∧ βˆ€π‘ ∈ (Filβ€˜π΅)(𝑐 ∈ (CauFiluβ€˜π‘ˆ) β†’ (𝐽 fLim 𝑐) β‰  βˆ…)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆ…c0 4322  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  TopOpenctopn 17369  Filcfil 23356   fLim cflim 23445  UnifStcuss 23765  UnifSpcusp 23766  CauFiluccfilu 23798  CUnifSpccusp 23809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7414  df-cusp 23810
This theorem is referenced by:  cmetcusp1  24877
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