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Theorem iscusp2 20931
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 20928 . 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 5875 . . . 4  |-  ( Fil `  B )  =  ( Fil `  ( Base `  W ) )
4 iscusp2.2 . . . . . . 7  |-  U  =  (UnifSt `  W )
54fveq2i 5875 . . . . . 6  |-  (CauFilu `  U
)  =  (CauFilu `  (UnifSt `  W ) )
65eleq2i 2535 . . . . 5  |-  ( c  e.  (CauFilu `  U )  <->  c  e.  (CauFilu `  (UnifSt `  W )
) )
7 iscusp2.3 . . . . . . 7  |-  J  =  ( TopOpen `  W )
87oveq1i 6306 . . . . . 6  |-  ( J 
fLim  c )  =  ( ( TopOpen `  W
)  fLim  c )
98neeq1i 2742 . . . . 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 2902 . . 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   (/)c0 3793   ` cfv 5594  (class class class)co 6296   Basecbs 14644   TopOpenctopn 14839   Filcfil 20472    fLim cflim 20561  UnifStcuss 20882  UnifSpcusp 20883  CauFiluccfilu 20915  CUnifSpccusp 20926
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 20927
This theorem is referenced by:  cmetcusp1OLD  21917  cmetcusp1  21918
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