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Theorem iscusp 23804
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 6897 . . 3 (𝑀 = π‘Š β†’ (Filβ€˜(Baseβ€˜π‘€)) = (Filβ€˜(Baseβ€˜π‘Š)))
2 2fveq3 6897 . . . . 5 (𝑀 = π‘Š β†’ (CauFiluβ€˜(UnifStβ€˜π‘€)) = (CauFiluβ€˜(UnifStβ€˜π‘Š)))
32eleq2d 2820 . . . 4 (𝑀 = π‘Š β†’ (𝑐 ∈ (CauFiluβ€˜(UnifStβ€˜π‘€)) ↔ 𝑐 ∈ (CauFiluβ€˜(UnifStβ€˜π‘Š))))
4 fveq2 6892 . . . . . 6 (𝑀 = π‘Š β†’ (TopOpenβ€˜π‘€) = (TopOpenβ€˜π‘Š))
54oveq1d 7424 . . . . 5 (𝑀 = π‘Š β†’ ((TopOpenβ€˜π‘€) fLim 𝑐) = ((TopOpenβ€˜π‘Š) fLim 𝑐))
65neeq1d 3001 . . . 4 (𝑀 = π‘Š β†’ (((TopOpenβ€˜π‘€) fLim 𝑐) β‰  βˆ… ↔ ((TopOpenβ€˜π‘Š) fLim 𝑐) β‰  βˆ…))
73, 6imbi12d 345 . . 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 3343 . 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 23803 . 2 CUnifSp = {𝑀 ∈ UnifSp ∣ βˆ€π‘ ∈ (Filβ€˜(Baseβ€˜π‘€))(𝑐 ∈ (CauFiluβ€˜(UnifStβ€˜π‘€)) β†’ ((TopOpenβ€˜π‘€) fLim 𝑐) β‰  βˆ…)}
108, 9elrab2 3687 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆ…c0 4323  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  TopOpenctopn 17367  Filcfil 23349   fLim cflim 23438  UnifStcuss 23758  UnifSpcusp 23759  CauFiluccfilu 23791  CUnifSpccusp 23802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-cusp 23803
This theorem is referenced by:  cuspusp  23805  cuspcvg  23806  iscusp2  23807  cmetcusp  24871
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