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Theorem usgfisALT 39866
Description: An undirected simple graph of finite order (i.e. with a finite number of vertices) is of finite size, i.e. it has also only a finite number of edges, analogous to usgrafis 25155. Remark: The proof of this theorem is very long compared with usgrafis 25155, because the theorem brfi1ind 12659 to perform the finite induction is taylored for binary relations, so that the theorem itself and the used lemmas must be transformed accordingly. Maybe a variant of brfi1ind 12659 could be provided, which is better suitable for this theorem. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Revised by AV, 15-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
usgfisALT  |-  ( G  e. FinUSGrph  ->  ( Edges  `  G )  e.  Fin )

Proof of Theorem usgfisALT
Dummy variables  e 
f  v  w  y  p  q  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fusgraimpclALT2 39847 . 2  |-  ( G  e. FinUSGrph  ->  ( G  e. USGrph  /\  ( 1st `  G
)  e.  Fin )
)
2 relusgra 25074 . . . . . . 7  |-  Rel USGrph
3 1st2nd 6844 . . . . . . 7  |-  ( ( Rel USGrph  /\  G  e. USGrph  )  ->  G  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
42, 3mpan 677 . . . . . 6  |-  ( G  e. USGrph  ->  G  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
5 eleq1 2519 . . . . . . . 8  |-  ( G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  ->  ( G  e. USGrph  <->  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  e. USGrph  )
)
6 df-br 4406 . . . . . . . 8  |-  ( ( 1st `  G ) USGrph 
( 2nd `  G
)  <->  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  e. USGrph  )
75, 6syl6bbr 267 . . . . . . 7  |-  ( G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  ->  ( G  e. USGrph  <->  ( 1st `  G
) USGrph  ( 2nd `  G
) ) )
8 mptresid 5162 . . . . . . . . . 10  |-  ( q  e.  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }  |->  q )  =  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e
>. )  |  n  e/  p } )
9 fvex 5880 . . . . . . . . . . 11  |-  ( Edges  `  <. v ,  e >. )  e.  _V
109mptrabex 6142 . . . . . . . . . 10  |-  ( q  e.  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }  |->  q )  e.  _V
118, 10eqeltrri 2528 . . . . . . . . 9  |-  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e
>. )  |  n  e/  p } )  e. 
_V
12 eleq1 2519 . . . . . . . . . 10  |-  ( e  =  ( 2nd `  G
)  ->  ( e  e.  Fin  <->  ( 2nd `  G
)  e.  Fin )
)
1312adantl 468 . . . . . . . . 9  |-  ( ( v  =  ( 1st `  G )  /\  e  =  ( 2nd `  G
) )  ->  (
e  e.  Fin  <->  ( 2nd `  G )  e.  Fin ) )
14 eleq1 2519 . . . . . . . . . 10  |-  ( e  =  f  ->  (
e  e.  Fin  <->  f  e.  Fin ) )
1514adantl 468 . . . . . . . . 9  |-  ( ( v  =  w  /\  e  =  f )  ->  ( e  e.  Fin  <->  f  e.  Fin ) )
16 df-br 4406 . . . . . . . . . 10  |-  ( v USGrph 
e  <->  <. v ,  e
>.  e. USGrph  )
17 vex 3050 . . . . . . . . . . . . 13  |-  v  e. 
_V
18 vex 3050 . . . . . . . . . . . . 13  |-  e  e. 
_V
1917, 18op1st 6806 . . . . . . . . . . . 12  |-  ( 1st `  <. v ,  e
>. )  =  v
2019eqcomi 2462 . . . . . . . . . . 11  |-  v  =  ( 1st `  <. v ,  e >. )
21 eqid 2453 . . . . . . . . . . 11  |-  ( Edges  `  <. v ,  e >. )  =  ( Edges  `  <. v ,  e >. )
22 eqid 2453 . . . . . . . . . . 11  |-  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }  =  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }
2320, 21, 22usgresvm1ALT 39863 . . . . . . . . . 10  |-  ( (
<. v ,  e >.  e. USGrph  /\  n  e.  v )  ->  ( v  \  { n } ) USGrph 
(  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } ) )
2416, 23sylanb 475 . . . . . . . . 9  |-  ( ( v USGrph  e  /\  n  e.  v )  ->  (
v  \  { n } ) USGrph  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } ) )
25 eleq1 2519 . . . . . . . . . 10  |-  ( f  =  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } )  ->  (
f  e.  Fin  <->  (  _I  |` 
{ p  e.  ( Edges  `  <. v ,  e
>. )  |  n  e/  p } )  e. 
Fin ) )
2625adantl 468 . . . . . . . . 9  |-  ( ( w  =  ( v 
\  { n }
)  /\  f  =  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } ) )  -> 
( f  e.  Fin  <->  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } )  e.  Fin ) )
27 usgrafisbaseALT 39856 . . . . . . . . 9  |-  ( ( v USGrph  e  /\  ( # `
 v )  =  0 )  ->  e  e.  Fin )
28 residfi 39049 . . . . . . . . . . . 12  |-  ( (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } )  e.  Fin  <->  {
p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }  e.  Fin )
29 simpr1 1015 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
v USGrph  e )
30 eleq1 2519 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( y  +  1 )  =  ( # `  v
)  ->  ( (
y  +  1 )  e.  NN0  <->  ( # `  v
)  e.  NN0 )
)
3130eqcoms 2461 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  v )  =  ( y  +  1 )  ->  (
( y  +  1 )  e.  NN0  <->  ( # `  v
)  e.  NN0 )
)
32 hashclb 12547 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( v  e.  _V  ->  (
v  e.  Fin  <->  ( # `  v
)  e.  NN0 )
)
3332bicomd 205 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( v  e.  _V  ->  (
( # `  v )  e.  NN0  <->  v  e.  Fin ) )
3417, 33ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  v )  e.  NN0  <->  v  e.  Fin )
3534biimpi 198 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  v )  e.  NN0  ->  v  e.  Fin )
3631, 35syl6bi 232 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  v )  =  ( y  +  1 )  ->  (
( y  +  1 )  e.  NN0  ->  v  e.  Fin ) )
37363ad2ant2 1031 . . . . . . . . . . . . . . . . . 18  |-  ( ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v )  ->  (
( y  +  1 )  e.  NN0  ->  v  e.  Fin ) )
3837impcom 432 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
v  e.  Fin )
3929, 38jca 535 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
( v USGrph  e  /\  v  e.  Fin )
)
40 usgrav 25077 . . . . . . . . . . . . . . . . . . 19  |-  ( v USGrph 
e  ->  ( v  e.  _V  /\  e  e. 
_V ) )
41 df-br 4406 . . . . . . . . . . . . . . . . . . . 20  |-  ( v FinUSGrph  e 
<-> 
<. v ,  e >.  e. FinUSGrph  )
42 isfusgra0 39841 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( v  e.  _V  /\  e  e.  _V )  ->  ( v FinUSGrph  e  <->  ( v USGrph  e  /\  v  e.  Fin ) ) )
4341, 42syl5bbr 263 . . . . . . . . . . . . . . . . . . 19  |-  ( ( v  e.  _V  /\  e  e.  _V )  ->  ( <. v ,  e
>.  e. FinUSGrph 
<->  ( v USGrph  e  /\  v  e.  Fin )
) )
4440, 43syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( v USGrph 
e  ->  ( <. v ,  e >.  e. FinUSGrph  <->  ( v USGrph  e  /\  v  e.  Fin ) ) )
45443ad2ant1 1030 . . . . . . . . . . . . . . . . 17  |-  ( ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v )  ->  ( <. v ,  e >.  e. FinUSGrph  <->  ( v USGrph  e  /\  v  e.  Fin ) ) )
4645adantl 468 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
( <. v ,  e
>.  e. FinUSGrph 
<->  ( v USGrph  e  /\  v  e.  Fin )
) )
4739, 46mpbird 236 . . . . . . . . . . . . . . 15  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  ->  <. v ,  e >.  e. FinUSGrph  )
48 eqidd 2454 . . . . . . . . . . . . . . . . . . . . 21  |-  ( v USGrph 
e  ->  ( 1st ` 
<. v ,  e >.
)  =  ( 1st `  <. v ,  e
>. ) )
4948, 19syl6req 2504 . . . . . . . . . . . . . . . . . . . 20  |-  ( v USGrph 
e  ->  v  =  ( 1st `  <. v ,  e >. )
)
5049eleq2d 2516 . . . . . . . . . . . . . . . . . . 19  |-  ( v USGrph 
e  ->  ( n  e.  v  <->  n  e.  ( 1st `  <. v ,  e
>. ) ) )
5150biimpd 211 . . . . . . . . . . . . . . . . . 18  |-  ( v USGrph 
e  ->  ( n  e.  v  ->  n  e.  ( 1st `  <. v ,  e >. )
) )
5251a1d 26 . . . . . . . . . . . . . . . . 17  |-  ( v USGrph 
e  ->  ( ( # `
 v )  =  ( y  +  1 )  ->  ( n  e.  v  ->  n  e.  ( 1st `  <. v ,  e >. )
) ) )
53523imp 1203 . . . . . . . . . . . . . . . 16  |-  ( ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v )  ->  n  e.  ( 1st `  <. v ,  e >. )
)
5453adantl 468 . . . . . . . . . . . . . . 15  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  ->  n  e.  ( 1st ` 
<. v ,  e >.
) )
5547, 54jca 535 . . . . . . . . . . . . . 14  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
( <. v ,  e
>.  e. FinUSGrph  /\  n  e.  ( 1st `  <. v ,  e >. )
) )
56 eqid 2453 . . . . . . . . . . . . . . 15  |-  ( 1st `  <. v ,  e
>. )  =  ( 1st `  <. v ,  e
>. )
57 df-ov 6298 . . . . . . . . . . . . . . 15  |-  ( v Edges 
e )  =  ( Edges  `  <. v ,  e
>. )
5857eqcomi 2462 . . . . . . . . . . . . . . . 16  |-  ( Edges  `  <. v ,  e >. )  =  ( v Edges  e
)
59 rabeq 3040 . . . . . . . . . . . . . . . 16  |-  ( ( Edges  `  <. v ,  e
>. )  =  (
v Edges  e )  ->  { p  e.  ( Edges  ` 
<. v ,  e >.
)  |  n  e/  p }  =  {
p  e.  ( v Edges 
e )  |  n  e/  p } )
6058, 59ax-mp 5 . . . . . . . . . . . . . . 15  |-  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }  =  { p  e.  ( v Edges  e )  |  n  e/  p }
6156, 57, 60usgfisALTlem2 39865 . . . . . . . . . . . . . 14  |-  ( (
<. v ,  e >.  e. FinUSGrph 
/\  n  e.  ( 1st `  <. v ,  e >. )
)  ->  ( (
v Edges  e )  e. 
Fin 
<->  { p  e.  ( Edges  `  <. v ,  e
>. )  |  n  e/  p }  e.  Fin ) )
6255, 61syl 17 . . . . . . . . . . . . 13  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
( ( v Edges  e
)  e.  Fin  <->  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }  e.  Fin )
)
6362biimprd 227 . . . . . . . . . . . 12  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
( { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p }  e.  Fin  ->  ( v Edges  e
)  e.  Fin )
)
6428, 63syl5bi 221 . . . . . . . . . . 11  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
( (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } )  e.  Fin  ->  ( v Edges  e )  e.  Fin ) )
6564imp 431 . . . . . . . . . 10  |-  ( ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `  v )  =  ( y  +  1 )  /\  n  e.  v ) )  /\  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } )  e.  Fin )  ->  ( v Edges  e
)  e.  Fin )
66 usgedgffibi 39850 . . . . . . . . . . . . 13  |-  ( v USGrph 
e  ->  ( e  e.  Fin  <->  ( v Edges  e
)  e.  Fin )
)
67663ad2ant1 1030 . . . . . . . . . . . 12  |-  ( ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v )  ->  (
e  e.  Fin  <->  ( v Edges  e )  e.  Fin ) )
6867adantl 468 . . . . . . . . . . 11  |-  ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `
 v )  =  ( y  +  1 )  /\  n  e.  v ) )  -> 
( e  e.  Fin  <->  (
v Edges  e )  e. 
Fin ) )
6968adantr 467 . . . . . . . . . 10  |-  ( ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `  v )  =  ( y  +  1 )  /\  n  e.  v ) )  /\  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } )  e.  Fin )  ->  ( e  e. 
Fin 
<->  ( v Edges  e )  e.  Fin ) )
7065, 69mpbird 236 . . . . . . . . 9  |-  ( ( ( ( y  +  1 )  e.  NN0  /\  ( v USGrph  e  /\  ( # `  v )  =  ( y  +  1 )  /\  n  e.  v ) )  /\  (  _I  |`  { p  e.  ( Edges  `  <. v ,  e >. )  |  n  e/  p } )  e.  Fin )  ->  e  e.  Fin )
712, 11, 13, 15, 24, 26, 27, 70brfi1ind 12659 . . . . . . . 8  |-  ( ( ( 1st `  G
) USGrph  ( 2nd `  G
)  /\  ( 1st `  G )  e.  Fin )  ->  ( 2nd `  G
)  e.  Fin )
7271ex 436 . . . . . . 7  |-  ( ( 1st `  G ) USGrph 
( 2nd `  G
)  ->  ( ( 1st `  G )  e. 
Fin  ->  ( 2nd `  G
)  e.  Fin )
)
737, 72syl6bi 232 . . . . . 6  |-  ( G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  ->  ( G  e. USGrph  ->  ( ( 1st `  G )  e.  Fin  ->  ( 2nd `  G )  e. 
Fin ) ) )
744, 73mpcom 37 . . . . 5  |-  ( G  e. USGrph  ->  ( ( 1st `  G )  e.  Fin  ->  ( 2nd `  G
)  e.  Fin )
)
7574imp 431 . . . 4  |-  ( ( G  e. USGrph  /\  ( 1st `  G )  e. 
Fin )  ->  ( 2nd `  G )  e. 
Fin )
76 rnfi 7862 . . . 4  |-  ( ( 2nd `  G )  e.  Fin  ->  ran  ( 2nd `  G )  e.  Fin )
7775, 76syl 17 . . 3  |-  ( ( G  e. USGrph  /\  ( 1st `  G )  e. 
Fin )  ->  ran  ( 2nd `  G )  e.  Fin )
78 edgval 25078 . . . . 5  |-  ( G  e. USGrph  ->  ( Edges  `  G )  =  ran  ( 2nd `  G ) )
7978eleq1d 2515 . . . 4  |-  ( G  e. USGrph  ->  ( ( Edges  `  G
)  e.  Fin  <->  ran  ( 2nd `  G )  e.  Fin ) )
8079adantr 467 . . 3  |-  ( ( G  e. USGrph  /\  ( 1st `  G )  e. 
Fin )  ->  (
( Edges  `  G )  e. 
Fin 
<->  ran  ( 2nd `  G
)  e.  Fin )
)
8177, 80mpbird 236 . 2  |-  ( ( G  e. USGrph  /\  ( 1st `  G )  e. 
Fin )  ->  ( Edges  `  G )  e.  Fin )
821, 81syl 17 1  |-  ( G  e. FinUSGrph  ->  ( Edges  `  G )  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    e/ wnel 2625   {crab 2743   _Vcvv 3047    \ cdif 3403   {csn 3970   <.cop 3976   class class class wbr 4405    |-> cmpt 4464    _I cid 4747   ran crn 4838    |` cres 4839   Rel wrel 4842   ` cfv 5585  (class class class)co 6295   1stc1st 6796   2ndc2nd 6797   Fincfn 7574   1c1 9545    + caddc 9547   NN0cn0 10876   #chash 12522   USGrph cusg 25069   Edges cedg 25070   FinUSGrph cfusg 39837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-hash 12523  df-uhgra 25031  df-umgra 25052  df-uslgra 25071  df-usgra 25072  df-edg 25075  df-gord 39802  df-gsiz 39803  df-fusg 39838
This theorem is referenced by: (None)
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