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Theorem iscusp 24024
Description: The predicate "π‘Š is a complete uniform space." (Contributed by Thierry Arnoux, 3-Dec-2017.)
Assertion
Ref Expression
iscusp (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…)))
Distinct variable group:   π‘Š,𝑐

Proof of Theorem iscusp
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 2fveq3 6896 . . 3 (𝑀 = π‘Š β†’ (Filβ€˜(Baseβ€˜π‘€)) = (Filβ€˜(Baseβ€˜π‘Š)))
2 2fveq3 6896 . . . . 5 (𝑀 = π‘Š β†’ (CauFiluβ€˜(UnifStβ€˜π‘€)) = (CauFiluβ€˜(UnifStβ€˜π‘Š)))
32eleq2d 2819 . . . 4 (𝑀 = π‘Š β†’ (𝑐 ∈ (CauFiluβ€˜(UnifStβ€˜π‘€)) ↔ 𝑐 ∈ (CauFiluβ€˜(UnifStβ€˜π‘Š))))
4 fveq2 6891 . . . . . 6 (𝑀 = π‘Š β†’ (TopOpenβ€˜π‘€) = (TopOpenβ€˜π‘Š))
54oveq1d 7426 . . . . 5 (𝑀 = π‘Š β†’ ((TopOpenβ€˜π‘€) fLim 𝑐) = ((TopOpenβ€˜π‘Š) fLim 𝑐))
65neeq1d 3000 . . . 4 (𝑀 = π‘Š β†’ (((TopOpenβ€˜π‘€) fLim 𝑐) β‰  βˆ… ↔ ((TopOpenβ€˜π‘Š) fLim 𝑐) β‰  βˆ…))
73, 6imbi12d 344 . . 3 (𝑀 = π‘Š β†’ ((𝑐 ∈ (CauFiluβ€˜(UnifStβ€˜π‘€)) β†’ ((TopOpenβ€˜π‘€) fLim 𝑐) β‰  βˆ…) ↔ (𝑐 ∈ (CauFiluβ€˜(UnifStβ€˜π‘Š)) β†’ ((TopOpenβ€˜π‘Š) fLim 𝑐) β‰  βˆ…)))
81, 7raleqbidv 3342 . 2 (𝑀 = π‘Š β†’ (βˆ€π‘ ∈ (Filβ€˜(Baseβ€˜π‘€))(𝑐 ∈ (CauFiluβ€˜(UnifStβ€˜π‘€)) β†’ ((TopOpenβ€˜π‘€) fLim 𝑐) β‰  βˆ…) ↔ βˆ€π‘ ∈ (Filβ€˜(Baseβ€˜π‘Š))(𝑐 ∈ (CauFiluβ€˜(UnifStβ€˜π‘Š)) β†’ ((TopOpenβ€˜π‘Š) fLim 𝑐) β‰  βˆ…)))
9 df-cusp 24023 . 2 CUnifSp = {𝑀 ∈ UnifSp ∣ βˆ€π‘ ∈ (Filβ€˜(Baseβ€˜π‘€))(𝑐 ∈ (CauFiluβ€˜(UnifStβ€˜π‘€)) β†’ ((TopOpenβ€˜π‘€) fLim 𝑐) β‰  βˆ…)}
108, 9elrab2 3686 1 (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…)))
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 17148  TopOpenctopn 17371  Filcfil 23569   fLim cflim 23658  UnifStcuss 23978  UnifSpcusp 23979  CauFiluccfilu 24011  CUnifSpccusp 24022
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 24023
This theorem is referenced by:  cuspusp  24025  cuspcvg  24026  iscusp2  24027  cmetcusp  25095
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