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Theorem iscusp 19877
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 5694 . . . 4  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
21fveq2d 5698 . . 3  |-  ( w  =  W  ->  ( Fil `  ( Base `  w
) )  =  ( Fil `  ( Base `  W ) ) )
3 fveq2 5694 . . . . . 6  |-  ( w  =  W  ->  (UnifSt `  w )  =  (UnifSt `  W ) )
43fveq2d 5698 . . . . 5  |-  ( w  =  W  ->  (CauFilu `  (UnifSt `  w ) )  =  (CauFilu `  (UnifSt `  W
) ) )
54eleq2d 2510 . . . 4  |-  ( w  =  W  ->  (
c  e.  (CauFilu `  (UnifSt `  w ) )  <->  c  e.  (CauFilu `  (UnifSt `  W )
) ) )
6 fveq2 5694 . . . . . 6  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  ( TopOpen `  W )
)
76oveq1d 6109 . . . . 5  |-  ( w  =  W  ->  (
( TopOpen `  w )  fLim  c )  =  ( ( TopOpen `  W )  fLim  c ) )
87neeq1d 2624 . . . 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 2934 . 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 19876 . 2  |- CUnifSp  =  {
w  e. UnifSp  |  A. c  e.  ( Fil `  ( Base `  w
) ) ( c  e.  (CauFilu `  (UnifSt `  w
) )  ->  (
( TopOpen `  w )  fLim  c )  =/=  (/) ) }
1210, 11elrab2 3122 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 1369    e. wcel 1756    =/= wne 2609   A.wral 2718   (/)c0 3640   ` cfv 5421  (class class class)co 6094   Basecbs 14177   TopOpenctopn 14363   Filcfil 19421    fLim cflim 19510  UnifStcuss 19831  UnifSpcusp 19832  CauFiluccfilu 19864  CUnifSpccusp 19875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-iota 5384  df-fv 5429  df-ov 6097  df-cusp 19876
This theorem is referenced by:  cuspusp  19878  cuspcvg  19879  iscusp2  19880  cmetcuspOLD  20868  cmetcusp  20869
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