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Theorem fin23lem27 8101
Description: The mapping constructed in fin23lem22 8100 is in fact an isomorphism. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypothesis
Ref Expression
fin23lem22.b  |-  C  =  ( i  e.  om  |->  ( iota_ j  e.  S
( j  i^i  S
)  ~~  i )
)
Assertion
Ref Expression
fin23lem27  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C  Isom  _E  ,  _E  ( om ,  S ) )
Distinct variable group:    i, j, S
Allowed substitution hints:    C( i, j)

Proof of Theorem fin23lem27
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordom 4768 . . . 4  |-  Ord  om
2 ordwe 4508 . . . 4  |-  ( Ord 
om  ->  _E  We  om )
3 weso 4487 . . . 4  |-  (  _E  We  om  ->  _E  Or  om )
41, 2, 3mp2b 9 . . 3  |-  _E  Or  om
54a1i 10 . 2  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  _E  Or  om )
6 sopo 4434 . . . . 5  |-  (  _E  Or  om  ->  _E  Po  om )
74, 6ax-mp 8 . . . 4  |-  _E  Po  om
8 poss 4419 . . . 4  |-  ( S 
C_  om  ->  (  _E  Po  om  ->  _E  Po  S ) )
97, 8mpi 16 . . 3  |-  ( S 
C_  om  ->  _E  Po  S )
109adantr 451 . 2  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  _E  Po  S )
11 fin23lem22.b . . . 4  |-  C  =  ( i  e.  om  |->  ( iota_ j  e.  S
( j  i^i  S
)  ~~  i )
)
1211fin23lem22 8100 . . 3  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C : om -1-1-onto-> S )
13 f1ofo 5585 . . 3  |-  ( C : om -1-1-onto-> S  ->  C : om -onto-> S )
1412, 13syl 15 . 2  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C : om -onto-> S
)
15 nnsdomel 7770 . . . . . . . 8  |-  ( ( a  e.  om  /\  b  e.  om )  ->  ( a  e.  b  <-> 
a  ~<  b ) )
1615adantl 452 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  <->  a  ~<  b ) )
1716biimpd 198 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  -> 
a  ~<  b ) )
18 fin23lem23 8099 . . . . . . . . . . . . 13  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  a
)
1918adantrr 697 . . . . . . . . . . . 12  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! j  e.  S  (
j  i^i  S )  ~~  a )
20 ineq1 3451 . . . . . . . . . . . . . 14  |-  ( j  =  i  ->  (
j  i^i  S )  =  ( i  i^i 
S ) )
2120breq1d 4135 . . . . . . . . . . . . 13  |-  ( j  =  i  ->  (
( j  i^i  S
)  ~~  a  <->  ( i  i^i  S )  ~~  a
) )
2221cbvreuv 2851 . . . . . . . . . . . 12  |-  ( E! j  e.  S  ( j  i^i  S ) 
~~  a  <->  E! i  e.  S  ( i  i^i  S )  ~~  a
)
2319, 22sylib 188 . . . . . . . . . . 11  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! i  e.  S  (
i  i^i  S )  ~~  a )
24 nfv 1624 . . . . . . . . . . . 12  |-  F/ i ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  i^i  S )  ~~  a
2521cbvriotav 6458 . . . . . . . . . . . 12  |-  ( iota_ j  e.  S ( j  i^i  S )  ~~  a )  =  (
iota_ i  e.  S
( i  i^i  S
)  ~~  a )
26 ineq1 3451 . . . . . . . . . . . . 13  |-  ( i  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  ->  ( i  i^i  S )  =  ( ( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  i^i  S ) )
2726breq1d 4135 . . . . . . . . . . . 12  |-  ( i  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  ->  ( (
i  i^i  S )  ~~  a  <->  ( ( iota_ j  e.  S ( j  i^i  S )  ~~  a )  i^i  S
)  ~~  a )
)
2824, 25, 27riotaprop 6470 . . . . . . . . . . 11  |-  ( E! i  e.  S  ( i  i^i  S ) 
~~  a  ->  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  e.  S  /\  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  i^i  S )  ~~  a
) )
2923, 28syl 15 . . . . . . . . . 10  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  e.  S  /\  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  i^i  S )  ~~  a
) )
3029simprd 449 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  i^i  S )  ~~  a
)
3130adantrr 697 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
( ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  i^i  S )  ~~  a )
32 simprr 733 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
a  ~<  b )
33 fin23lem23 8099 . . . . . . . . . . . . . . 15  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  b  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  b
)
3433adantrl 696 . . . . . . . . . . . . . 14  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! j  e.  S  (
j  i^i  S )  ~~  b )
3520breq1d 4135 . . . . . . . . . . . . . . 15  |-  ( j  =  i  ->  (
( j  i^i  S
)  ~~  b  <->  ( i  i^i  S )  ~~  b
) )
3635cbvreuv 2851 . . . . . . . . . . . . . 14  |-  ( E! j  e.  S  ( j  i^i  S ) 
~~  b  <->  E! i  e.  S  ( i  i^i  S )  ~~  b
)
3734, 36sylib 188 . . . . . . . . . . . . 13  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! i  e.  S  (
i  i^i  S )  ~~  b )
38 nfv 1624 . . . . . . . . . . . . . 14  |-  F/ i ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  i^i  S )  ~~  b
3935cbvriotav 6458 . . . . . . . . . . . . . 14  |-  ( iota_ j  e.  S ( j  i^i  S )  ~~  b )  =  (
iota_ i  e.  S
( i  i^i  S
)  ~~  b )
40 ineq1 3451 . . . . . . . . . . . . . . 15  |-  ( i  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  ->  ( i  i^i  S )  =  ( ( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S ) )
4140breq1d 4135 . . . . . . . . . . . . . 14  |-  ( i  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  ->  ( (
i  i^i  S )  ~~  b  <->  ( ( iota_ j  e.  S ( j  i^i  S )  ~~  b )  i^i  S
)  ~~  b )
)
4238, 39, 41riotaprop 6470 . . . . . . . . . . . . 13  |-  ( E! i  e.  S  ( i  i^i  S ) 
~~  b  ->  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  e.  S  /\  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S )  ~~  b
) )
4337, 42syl 15 . . . . . . . . . . . 12  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  e.  S  /\  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S )  ~~  b
) )
4443simprd 449 . . . . . . . . . . 11  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S )  ~~  b
)
45 ensym 7053 . . . . . . . . . . 11  |-  ( ( ( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S )  ~~  b  ->  b  ~~  ( (
iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S ) )
4644, 45syl 15 . . . . . . . . . 10  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  b  ~~  ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  i^i  S )
)
4746adantrr 697 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
b  ~~  ( ( iota_ j  e.  S ( j  i^i  S ) 
~~  b )  i^i 
S ) )
48 sdomentr 7138 . . . . . . . . 9  |-  ( ( a  ~<  b  /\  b  ~~  ( ( iota_ j  e.  S ( j  i^i  S )  ~~  b )  i^i  S
) )  ->  a  ~<  ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  i^i  S )
)
4932, 47, 48syl2anc 642 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
a  ~<  ( ( iota_ j  e.  S ( j  i^i  S )  ~~  b )  i^i  S
) )
50 ensdomtr 7140 . . . . . . . 8  |-  ( ( ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  i^i  S )  ~~  a  /\  a  ~<  ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  i^i  S )
)  ->  ( ( iota_ j  e.  S ( j  i^i  S ) 
~~  a )  i^i 
S )  ~<  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S ) )
5131, 49, 50syl2anc 642 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
( ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  i^i  S )  ~<  ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  i^i  S )
)
5251expr 598 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  ~<  b  ->  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  i^i  S )  ~<  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S ) ) )
53 simpll 730 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  S  C_ 
om )
54 omsson 4763 . . . . . . . . 9  |-  om  C_  On
5553, 54syl6ss 3277 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  S  C_  On )
5629simpld 445 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  a )  e.  S )
5755, 56sseldd 3267 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  a )  e.  On )
5843simpld 445 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  b )  e.  S )
5955, 58sseldd 3267 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  b )  e.  On )
60 onsdominel 7153 . . . . . . . 8  |-  ( ( ( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  e.  On  /\  ( iota_ j  e.  S ( j  i^i  S )  ~~  b )  e.  On  /\  ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  i^i  S )  ~<  ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  i^i  S )
)  ->  ( iota_ j  e.  S ( j  i^i  S )  ~~  a )  e.  (
iota_ j  e.  S
( j  i^i  S
)  ~~  b )
)
61603expia 1154 . . . . . . 7  |-  ( ( ( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  e.  On  /\  ( iota_ j  e.  S ( j  i^i  S )  ~~  b )  e.  On )  ->  ( ( (
iota_ j  e.  S
( j  i^i  S
)  ~~  a )  i^i  S )  ~<  (
( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  i^i  S )  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  a )  e.  ( iota_ j  e.  S
( j  i^i  S
)  ~~  b )
) )
6257, 59, 61syl2anc 642 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
)  i^i  S )  ~<  ( ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
)  i^i  S )  ->  ( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  e.  ( iota_ j  e.  S
( j  i^i  S
)  ~~  b )
) )
6317, 52, 623syld 51 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  -> 
( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  e.  ( iota_ j  e.  S
( j  i^i  S
)  ~~  b )
) )
64 simprl 732 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  a  e.  om )
65 breq2 4129 . . . . . . . . 9  |-  ( i  =  a  ->  (
( j  i^i  S
)  ~~  i  <->  ( j  i^i  S )  ~~  a
) )
6665riotabidv 6448 . . . . . . . 8  |-  ( i  =  a  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  i )  =  ( iota_ j  e.  S
( j  i^i  S
)  ~~  a )
)
6766, 11fvmptg 5707 . . . . . . 7  |-  ( ( a  e.  om  /\  ( iota_ j  e.  S
( j  i^i  S
)  ~~  a )  e.  S )  ->  ( C `  a )  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
) )
6864, 56, 67syl2anc 642 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( C `  a )  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  a
) )
69 simprr 733 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  b  e.  om )
70 breq2 4129 . . . . . . . . 9  |-  ( i  =  b  ->  (
( j  i^i  S
)  ~~  i  <->  ( j  i^i  S )  ~~  b
) )
7170riotabidv 6448 . . . . . . . 8  |-  ( i  =  b  ->  ( iota_ j  e.  S ( j  i^i  S ) 
~~  i )  =  ( iota_ j  e.  S
( j  i^i  S
)  ~~  b )
)
7271, 11fvmptg 5707 . . . . . . 7  |-  ( ( b  e.  om  /\  ( iota_ j  e.  S
( j  i^i  S
)  ~~  b )  e.  S )  ->  ( C `  b )  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
) )
7369, 58, 72syl2anc 642 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( C `  b )  =  ( iota_ j  e.  S ( j  i^i 
S )  ~~  b
) )
7468, 73eleq12d 2434 . . . . 5  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
( C `  a
)  e.  ( C `
 b )  <->  ( iota_ j  e.  S ( j  i^i  S )  ~~  a )  e.  (
iota_ j  e.  S
( j  i^i  S
)  ~~  b )
) )
7563, 74sylibrd 225 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  -> 
( C `  a
)  e.  ( C `
 b ) ) )
76 epel 4411 . . . 4  |-  ( a  _E  b  <->  a  e.  b )
77 fvex 5646 . . . . 5  |-  ( C `
 b )  e. 
_V
7877epelc 4410 . . . 4  |-  ( ( C `  a )  _E  ( C `  b )  <->  ( C `  a )  e.  ( C `  b ) )
7975, 76, 783imtr4g 261 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  _E  b  -> 
( C `  a
)  _E  ( C `
 b ) ) )
8079ralrimivva 2720 . 2  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  A. a  e.  om  A. b  e.  om  (
a  _E  b  -> 
( C `  a
)  _E  ( C `
 b ) ) )
81 soisoi 5948 . 2  |-  ( ( (  _E  Or  om  /\  _E  Po  S )  /\  ( C : om -onto-> S  /\  A. a  e.  om  A. b  e. 
om  ( a  _E  b  ->  ( C `  a )  _E  ( C `  b )
) ) )  ->  C  Isom  _E  ,  _E  ( om ,  S ) )
825, 10, 14, 80, 81syl22anc 1184 1  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C  Isom  _E  ,  _E  ( om ,  S ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   A.wral 2628   E!wreu 2630    i^i cin 3237    C_ wss 3238   class class class wbr 4125    e. cmpt 4179    _E cep 4406    Po wpo 4415    Or wor 4416    We wwe 4454   Ord word 4494   Oncon0 4495   omcom 4759   -onto->wfo 5356   -1-1-onto->wf1o 5357   ` cfv 5358    Isom wiso 5359   iota_crio 6439    ~~ cen 7003    ~< csdm 7005   Fincfn 7006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-riota 6446  df-recs 6530  df-1o 6621  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-card 7719
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