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Theorem fin23lem27 8758
Description: The mapping constructed in fin23lem22 8757 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 6701 . . . 4  |-  Ord  om
2 ordwe 5436 . . . 4  |-  ( Ord 
om  ->  _E  We  om )
3 weso 4825 . . . 4  |-  (  _E  We  om  ->  _E  Or  om )
41, 2, 3mp2b 10 . . 3  |-  _E  Or  om
54a1i 11 . 2  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  _E  Or  om )
6 sopo 4772 . . . . 5  |-  (  _E  Or  om  ->  _E  Po  om )
74, 6ax-mp 5 . . . 4  |-  _E  Po  om
8 poss 4757 . . . 4  |-  ( S 
C_  om  ->  (  _E  Po  om  ->  _E  Po  S ) )
97, 8mpi 20 . . 3  |-  ( S 
C_  om  ->  _E  Po  S )
109adantr 467 . 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 8757 . . 3  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C : om -1-1-onto-> S )
13 f1ofo 5821 . . 3  |-  ( C : om -1-1-onto-> S  ->  C : om -onto-> S )
1412, 13syl 17 . 2  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C : om -onto-> S
)
15 nnsdomel 8424 . . . . . . . 8  |-  ( ( a  e.  om  /\  b  e.  om )  ->  ( a  e.  b  <-> 
a  ~<  b ) )
1615adantl 468 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  <->  a  ~<  b ) )
1716biimpd 211 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  -> 
a  ~<  b ) )
18 fin23lem23 8756 . . . . . . . . . . . . 13  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  a  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  a
)
1918adantrr 723 . . . . . . . . . . . 12  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! j  e.  S  (
j  i^i  S )  ~~  a )
20 ineq1 3627 . . . . . . . . . . . . . 14  |-  ( j  =  i  ->  (
j  i^i  S )  =  ( i  i^i 
S ) )
2120breq1d 4412 . . . . . . . . . . . . 13  |-  ( j  =  i  ->  (
( j  i^i  S
)  ~~  a  <->  ( i  i^i  S )  ~~  a
) )
2221cbvreuv 3021 . . . . . . . . . . . 12  |-  ( E! j  e.  S  ( j  i^i  S ) 
~~  a  <->  E! i  e.  S  ( i  i^i  S )  ~~  a
)
2319, 22sylib 200 . . . . . . . . . . 11  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! i  e.  S  (
i  i^i  S )  ~~  a )
24 nfv 1761 . . . . . . . . . . . 12  |-  F/ i ( ( iota_ j  e.  S  ( j  i^i 
S )  ~~  a
)  i^i  S )  ~~  a
2521cbvriotav 6263 . . . . . . . . . . . 12  |-  ( iota_ j  e.  S  ( j  i^i  S )  ~~  a )  =  (
iota_ i  e.  S  ( i  i^i  S
)  ~~  a )
26 ineq1 3627 . . . . . . . . . . . . 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 4412 . . . . . . . . . . . 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 6275 . . . . . . . . . . 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 17 . . . . . . . . . 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 465 . . . . . . . . 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 723 . . . . . . . 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 766 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( ( a  e.  om  /\  b  e.  om )  /\  a  ~<  b ) )  -> 
a  ~<  b )
33 fin23lem23 8756 . . . . . . . . . . . . . . 15  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  b  e.  om )  ->  E! j  e.  S  ( j  i^i 
S )  ~~  b
)
3433adantrl 722 . . . . . . . . . . . . . 14  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! j  e.  S  (
j  i^i  S )  ~~  b )
3520breq1d 4412 . . . . . . . . . . . . . . 15  |-  ( j  =  i  ->  (
( j  i^i  S
)  ~~  b  <->  ( i  i^i  S )  ~~  b
) )
3635cbvreuv 3021 . . . . . . . . . . . . . 14  |-  ( E! j  e.  S  ( j  i^i  S ) 
~~  b  <->  E! i  e.  S  ( i  i^i  S )  ~~  b
)
3734, 36sylib 200 . . . . . . . . . . . . 13  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  E! i  e.  S  (
i  i^i  S )  ~~  b )
38 nfv 1761 . . . . . . . . . . . . . 14  |-  F/ i ( ( iota_ j  e.  S  ( j  i^i 
S )  ~~  b
)  i^i  S )  ~~  b
3935cbvriotav 6263 . . . . . . . . . . . . . 14  |-  ( iota_ j  e.  S  ( j  i^i  S )  ~~  b )  =  (
iota_ i  e.  S  ( i  i^i  S
)  ~~  b )
40 ineq1 3627 . . . . . . . . . . . . . . 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 4412 . . . . . . . . . . . . . 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 6275 . . . . . . . . . . . . 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 17 . . . . . . . . . . . 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 465 . . . . . . . . . . 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
)
4544ensymd 7620 . . . . . . . . . 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 )
)
4645adantrr 723 . . . . . . . . 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 ) )
47 sdomentr 7706 . . . . . . . . 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 )
)
4832, 46, 47syl2anc 667 . . . . . . . 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
) )
49 ensdomtr 7708 . . . . . . . 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 ) )
5031, 48, 49syl2anc 667 . . . . . . 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 )
)
5150expr 620 . . . . . 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 ) ) )
52 simpll 760 . . . . . . . . 9  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  S  C_ 
om )
53 omsson 6696 . . . . . . . . 9  |-  om  C_  On
5452, 53syl6ss 3444 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  S  C_  On )
5529simpld 461 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S  ( j  i^i  S ) 
~~  a )  e.  S )
5654, 55sseldd 3433 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S  ( j  i^i  S ) 
~~  a )  e.  On )
5743simpld 461 . . . . . . . 8  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S  ( j  i^i  S ) 
~~  b )  e.  S )
5854, 57sseldd 3433 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( iota_ j  e.  S  ( j  i^i  S ) 
~~  b )  e.  On )
59 onsdominel 7721 . . . . . . . 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 )
)
60593expia 1210 . . . . . . 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 )
) )
6156, 58, 60syl2anc 667 . . . . . 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 )
) )
6217, 51, 613syld 57 . . . . 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 )
) )
63 simprl 764 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  a  e.  om )
64 breq2 4406 . . . . . . . . 9  |-  ( i  =  a  ->  (
( j  i^i  S
)  ~~  i  <->  ( j  i^i  S )  ~~  a
) )
6564riotabidv 6254 . . . . . . . 8  |-  ( i  =  a  ->  ( iota_ j  e.  S  ( j  i^i  S ) 
~~  i )  =  ( iota_ j  e.  S  ( j  i^i  S
)  ~~  a )
)
6665, 11fvmptg 5946 . . . . . . 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
) )
6763, 55, 66syl2anc 667 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( C `  a )  =  ( iota_ j  e.  S  ( j  i^i 
S )  ~~  a
) )
68 simprr 766 . . . . . . 7  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  b  e.  om )
69 breq2 4406 . . . . . . . . 9  |-  ( i  =  b  ->  (
( j  i^i  S
)  ~~  i  <->  ( j  i^i  S )  ~~  b
) )
7069riotabidv 6254 . . . . . . . 8  |-  ( i  =  b  ->  ( iota_ j  e.  S  ( j  i^i  S ) 
~~  i )  =  ( iota_ j  e.  S  ( j  i^i  S
)  ~~  b )
)
7170, 11fvmptg 5946 . . . . . . 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
) )
7268, 57, 71syl2anc 667 . . . . . 6  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  ( C `  b )  =  ( iota_ j  e.  S  ( j  i^i 
S )  ~~  b
) )
7367, 72eleq12d 2523 . . . . 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 )
) )
7462, 73sylibrd 238 . . . 4  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  e.  b  -> 
( C `  a
)  e.  ( C `
 b ) ) )
75 epel 4748 . . . 4  |-  ( a  _E  b  <->  a  e.  b )
76 fvex 5875 . . . . 5  |-  ( C `
 b )  e. 
_V
7776epelc 4747 . . . 4  |-  ( ( C `  a )  _E  ( C `  b )  <->  ( C `  a )  e.  ( C `  b ) )
7874, 75, 773imtr4g 274 . . 3  |-  ( ( ( S  C_  om  /\  -.  S  e.  Fin )  /\  ( a  e. 
om  /\  b  e.  om ) )  ->  (
a  _E  b  -> 
( C `  a
)  _E  ( C `
 b ) ) )
7978ralrimivva 2809 . 2  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  A. a  e.  om  A. b  e.  om  (
a  _E  b  -> 
( C `  a
)  _E  ( C `
 b ) ) )
80 soisoi 6219 . 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 ) )
815, 10, 14, 79, 80syl22anc 1269 1  |-  ( ( S  C_  om  /\  -.  S  e.  Fin )  ->  C  Isom  _E  ,  _E  ( om ,  S ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   E!wreu 2739    i^i cin 3403    C_ wss 3404   class class class wbr 4402    |-> cmpt 4461    _E cep 4743    Po wpo 4753    Or wor 4754    We wwe 4792   Ord word 5422   Oncon0 5423   -onto->wfo 5580   -1-1-onto->wf1o 5581   ` cfv 5582    Isom wiso 5583   iota_crio 6251   omcom 6692    ~~ cen 7566    ~< csdm 7568   Fincfn 7569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-om 6693  df-wrecs 7028  df-recs 7090  df-1o 7182  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373
This theorem is referenced by: (None)
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