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Theorem iscusp2 21366
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 21363 . 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 5891 . . . 4  |-  ( Fil `  B )  =  ( Fil `  ( Base `  W ) )
4 iscusp2.2 . . . . . . 7  |-  U  =  (UnifSt `  W )
54fveq2i 5891 . . . . . 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 6325 . . . . . 6  |-  ( J 
fLim  c )  =  ( ( TopOpen `  W
)  fLim  c )
98neeq1i 2700 . . . . 5  |-  ( ( J  fLim  c )  =/=  (/)  <->  ( ( TopOpen `  W )  fLim  c
)  =/=  (/) )
106, 9imbi12i 332 . . . 4  |-  ( ( c  e.  (CauFilu `  U
)  ->  ( J  fLim  c )  =/=  (/) )  <->  ( c  e.  (CauFilu `  (UnifSt `  W
) )  ->  (
( TopOpen `  W )  fLim  c )  =/=  (/) ) )
113, 10raleqbii 2845 . . 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 705 . 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 260 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 189    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   (/)c0 3743   ` cfv 5601  (class class class)co 6315   Basecbs 15170   TopOpenctopn 15369   Filcfil 20909    fLim cflim 20998  UnifStcuss 21317  UnifSpcusp 21318  CauFiluccfilu 21350  CUnifSpccusp 21361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-iota 5565  df-fv 5609  df-ov 6318  df-cusp 21362
This theorem is referenced by:  cmetcusp1  22369
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