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Theorem iscusp2 20012
Description: The predicate " W is a complete uniform space." (Contributed by Thierry Arnoux, 15-Dec-2017.)
Hypotheses
Ref Expression
iscusp2.1  |-  B  =  ( Base `  W
)
iscusp2.2  |-  U  =  (UnifSt `  W )
iscusp2.3  |-  J  =  ( TopOpen `  W )
Assertion
Ref Expression
iscusp2  |-  ( W  e. CUnifSp 
<->  ( W  e. UnifSp  /\  A. c  e.  ( Fil `  B ) ( c  e.  (CauFilu `  U )  -> 
( J  fLim  c
)  =/=  (/) ) ) )
Distinct variable group:    W, c
Allowed substitution hints:    B( c)    U( c)    J( c)

Proof of Theorem iscusp2
StepHypRef Expression
1 iscusp 20009 . 2  |-  ( W  e. CUnifSp 
<->  ( W  e. UnifSp  /\  A. c  e.  ( Fil `  ( Base `  W
) ) ( c  e.  (CauFilu `  (UnifSt `  W
) )  ->  (
( TopOpen `  W )  fLim  c )  =/=  (/) ) ) )
2 iscusp2.1 . . . . 5  |-  B  =  ( Base `  W
)
32fveq2i 5805 . . . 4  |-  ( Fil `  B )  =  ( Fil `  ( Base `  W ) )
4 iscusp2.2 . . . . . . 7  |-  U  =  (UnifSt `  W )
54fveq2i 5805 . . . . . 6  |-  (CauFilu `  U
)  =  (CauFilu `  (UnifSt `  W ) )
65eleq2i 2532 . . . . 5  |-  ( c  e.  (CauFilu `  U )  <->  c  e.  (CauFilu `  (UnifSt `  W )
) )
7 iscusp2.3 . . . . . . 7  |-  J  =  ( TopOpen `  W )
87oveq1i 6213 . . . . . 6  |-  ( J 
fLim  c )  =  ( ( TopOpen `  W
)  fLim  c )
98neeq1i 2737 . . . . 5  |-  ( ( J  fLim  c )  =/=  (/)  <->  ( ( TopOpen `  W )  fLim  c
)  =/=  (/) )
106, 9imbi12i 326 . . . 4  |-  ( ( c  e.  (CauFilu `  U
)  ->  ( J  fLim  c )  =/=  (/) )  <->  ( c  e.  (CauFilu `  (UnifSt `  W
) )  ->  (
( TopOpen `  W )  fLim  c )  =/=  (/) ) )
113, 10raleqbii 2848 . . 3  |-  ( A. c  e.  ( Fil `  B ) ( c  e.  (CauFilu `  U )  -> 
( J  fLim  c
)  =/=  (/) )  <->  A. c  e.  ( Fil `  ( Base `  W ) ) ( c  e.  (CauFilu `  (UnifSt `  W )
)  ->  ( ( TopOpen
`  W )  fLim  c )  =/=  (/) ) )
1211anbi2i 694 . 2  |-  ( ( W  e. UnifSp  /\  A. c  e.  ( Fil `  B
) ( c  e.  (CauFilu `  U )  -> 
( J  fLim  c
)  =/=  (/) ) )  <-> 
( W  e. UnifSp  /\  A. c  e.  ( Fil `  ( Base `  W
) ) ( c  e.  (CauFilu `  (UnifSt `  W
) )  ->  (
( TopOpen `  W )  fLim  c )  =/=  (/) ) ) )
131, 12bitr4i 252 1  |-  ( W  e. CUnifSp 
<->  ( W  e. UnifSp  /\  A. c  e.  ( Fil `  B ) ( c  e.  (CauFilu `  U )  -> 
( J  fLim  c
)  =/=  (/) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   (/)c0 3748   ` cfv 5529  (class class class)co 6203   Basecbs 14295   TopOpenctopn 14482   Filcfil 19553    fLim cflim 19642  UnifStcuss 19963  UnifSpcusp 19964  CauFiluccfilu 19996  CUnifSpccusp 20007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-ov 6206  df-cusp 20008
This theorem is referenced by:  cmetcusp1OLD  20998  cmetcusp1  20999
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