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Theorem iscusp 20927
Description: The predicate " W is a complete uniform space." (Contributed by Thierry Arnoux, 3-Dec-2017.)
Assertion
Ref Expression
iscusp  |-  ( W  e. CUnifSp 
<->  ( W  e. UnifSp  /\  A. c  e.  ( Fil `  ( Base `  W
) ) ( c  e.  (CauFilu `  (UnifSt `  W
) )  ->  (
( TopOpen `  W )  fLim  c )  =/=  (/) ) ) )
Distinct variable group:    W, c

Proof of Theorem iscusp
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . . 4  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
21fveq2d 5876 . . 3  |-  ( w  =  W  ->  ( Fil `  ( Base `  w
) )  =  ( Fil `  ( Base `  W ) ) )
3 fveq2 5872 . . . . . 6  |-  ( w  =  W  ->  (UnifSt `  w )  =  (UnifSt `  W ) )
43fveq2d 5876 . . . . 5  |-  ( w  =  W  ->  (CauFilu `  (UnifSt `  w ) )  =  (CauFilu `  (UnifSt `  W
) ) )
54eleq2d 2527 . . . 4  |-  ( w  =  W  ->  (
c  e.  (CauFilu `  (UnifSt `  w ) )  <->  c  e.  (CauFilu `  (UnifSt `  W )
) ) )
6 fveq2 5872 . . . . . 6  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  ( TopOpen `  W )
)
76oveq1d 6311 . . . . 5  |-  ( w  =  W  ->  (
( TopOpen `  w )  fLim  c )  =  ( ( TopOpen `  W )  fLim  c ) )
87neeq1d 2734 . . . 4  |-  ( w  =  W  ->  (
( ( TopOpen `  w
)  fLim  c )  =/=  (/)  <->  ( ( TopOpen `  W )  fLim  c
)  =/=  (/) ) )
95, 8imbi12d 320 . . 3  |-  ( w  =  W  ->  (
( c  e.  (CauFilu `  (UnifSt `  w )
)  ->  ( ( TopOpen
`  w )  fLim  c )  =/=  (/) )  <->  ( c  e.  (CauFilu `  (UnifSt `  W
) )  ->  (
( TopOpen `  W )  fLim  c )  =/=  (/) ) ) )
102, 9raleqbidv 3068 . 2  |-  ( w  =  W  ->  ( A. c  e.  ( Fil `  ( Base `  w
) ) ( c  e.  (CauFilu `  (UnifSt `  w
) )  ->  (
( TopOpen `  w )  fLim  c )  =/=  (/) )  <->  A. c  e.  ( Fil `  ( Base `  W ) ) ( c  e.  (CauFilu `  (UnifSt `  W )
)  ->  ( ( TopOpen
`  W )  fLim  c )  =/=  (/) ) ) )
11 df-cusp 20926 . 2  |- CUnifSp  =  {
w  e. UnifSp  |  A. c  e.  ( Fil `  ( Base `  w
) ) ( c  e.  (CauFilu `  (UnifSt `  w
) )  ->  (
( TopOpen `  w )  fLim  c )  =/=  (/) ) }
1210, 11elrab2 3259 1  |-  ( W  e. CUnifSp 
<->  ( W  e. UnifSp  /\  A. c  e.  ( Fil `  ( Base `  W
) ) ( c  e.  (CauFilu `  (UnifSt `  W
) )  ->  (
( TopOpen `  W )  fLim  c )  =/=  (/) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   (/)c0 3793   ` cfv 5594  (class class class)co 6296   Basecbs 14643   TopOpenctopn 14838   Filcfil 20471    fLim cflim 20560  UnifStcuss 20881  UnifSpcusp 20882  CauFiluccfilu 20914  CUnifSpccusp 20925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-cusp 20926
This theorem is referenced by:  cuspusp  20928  cuspcvg  20929  iscusp2  20930  cmetcuspOLD  21918  cmetcusp  21919
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