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Theorem iscusp 19833
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 5688 . . . 4  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
21fveq2d 5692 . . 3  |-  ( w  =  W  ->  ( Fil `  ( Base `  w
) )  =  ( Fil `  ( Base `  W ) ) )
3 fveq2 5688 . . . . . 6  |-  ( w  =  W  ->  (UnifSt `  w )  =  (UnifSt `  W ) )
43fveq2d 5692 . . . . 5  |-  ( w  =  W  ->  (CauFilu `  (UnifSt `  w ) )  =  (CauFilu `  (UnifSt `  W
) ) )
54eleq2d 2508 . . . 4  |-  ( w  =  W  ->  (
c  e.  (CauFilu `  (UnifSt `  w ) )  <->  c  e.  (CauFilu `  (UnifSt `  W )
) ) )
6 fveq2 5688 . . . . . 6  |-  ( w  =  W  ->  ( TopOpen
`  w )  =  ( TopOpen `  W )
)
76oveq1d 6105 . . . . 5  |-  ( w  =  W  ->  (
( TopOpen `  w )  fLim  c )  =  ( ( TopOpen `  W )  fLim  c ) )
87neeq1d 2619 . . . 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 2929 . 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 19832 . 2  |- CUnifSp  =  {
w  e. UnifSp  |  A. c  e.  ( Fil `  ( Base `  w
) ) ( c  e.  (CauFilu `  (UnifSt `  w
) )  ->  (
( TopOpen `  w )  fLim  c )  =/=  (/) ) }
1210, 11elrab2 3116 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 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   (/)c0 3634   ` cfv 5415  (class class class)co 6090   Basecbs 14170   TopOpenctopn 14356   Filcfil 19377    fLim cflim 19466  UnifStcuss 19787  UnifSpcusp 19788  CauFiluccfilu 19820  CUnifSpccusp 19831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-iota 5378  df-fv 5423  df-ov 6093  df-cusp 19832
This theorem is referenced by:  cuspusp  19834  cuspcvg  19835  iscusp2  19836  cmetcuspOLD  20824  cmetcusp  20825
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