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Theorem dfrtrcl2 12920
Description: The two definitions  t* and  t*rec of the reflexive, transitive closure coincide if  R is indeed a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
Hypotheses
Ref Expression
drrtrcl2.1  |-  ( ph  ->  Rel  R )
drrtrcl2.2  |-  ( ph  ->  R  e.  _V )
Assertion
Ref Expression
dfrtrcl2  |-  ( ph  ->  ( t* `  R )  =  ( t*rec `  R
) )

Proof of Theorem dfrtrcl2
Dummy variables  x  z  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2397 . . . 4  |-  ( ph  ->  ( x  e.  _V  |->  |^|
{ z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } )  =  ( x  e.  _V  |->  |^|
{ z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } ) )
2 dmeq 5133 . . . . . . . . . . 11  |-  ( x  =  R  ->  dom  x  =  dom  R )
3 rneq 5158 . . . . . . . . . . 11  |-  ( x  =  R  ->  ran  x  =  ran  R )
42, 3uneq12d 3590 . . . . . . . . . 10  |-  ( x  =  R  ->  ( dom  x  u.  ran  x
)  =  ( dom 
R  u.  ran  R
) )
54reseq2d 5203 . . . . . . . . 9  |-  ( x  =  R  ->  (  _I  |`  ( dom  x  u.  ran  x ) )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
65sseq1d 3461 . . . . . . . 8  |-  ( x  =  R  ->  (
(  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  <->  (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z )
)
7 id 22 . . . . . . . . 9  |-  ( x  =  R  ->  x  =  R )
87sseq1d 3461 . . . . . . . 8  |-  ( x  =  R  ->  (
x  C_  z  <->  R  C_  z
) )
96, 83anbi12d 1298 . . . . . . 7  |-  ( x  =  R  ->  (
( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z )  <->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z )
) )
109abbidv 2532 . . . . . 6  |-  ( x  =  R  ->  { z  |  ( (  _I  |`  ( dom  x  u. 
ran  x ) ) 
C_  z  /\  x  C_  z  /\  ( z  o.  z )  C_  z ) }  =  { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } )
1110inteqd 4221 . . . . 5  |-  ( x  =  R  ->  |^| { z  |  ( (  _I  |`  ( dom  x  u. 
ran  x ) ) 
C_  z  /\  x  C_  z  /\  ( z  o.  z )  C_  z ) }  =  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } )
1211adantl 464 . . . 4  |-  ( (
ph  /\  x  =  R )  ->  |^| { z  |  ( (  _I  |`  ( dom  x  u. 
ran  x ) ) 
C_  z  /\  x  C_  z  /\  ( z  o.  z )  C_  z ) }  =  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } )
13 drrtrcl2.2 . . . 4  |-  ( ph  ->  R  e.  _V )
14 drrtrcl2.1 . . . . . . . . . 10  |-  ( ph  ->  Rel  R )
15 relfld 5458 . . . . . . . . . 10  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
1614, 15syl 16 . . . . . . . . 9  |-  ( ph  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
1716eqcomd 2404 . . . . . . . 8  |-  ( ph  ->  ( dom  R  u.  ran  R )  =  U. U. R )
1814, 13rtrclreclem1 12916 . . . . . . . . 9  |-  ( ph  ->  (  _I  |`  U. U. R )  C_  (
t*rec `  R
) )
19 id 22 . . . . . . . . . . 11  |-  ( ( dom  R  u.  ran  R )  =  U. U. R  ->  ( dom  R  u.  ran  R )  = 
U. U. R )
2019reseq2d 5203 . . . . . . . . . 10  |-  ( ( dom  R  u.  ran  R )  =  U. U. R  ->  (  _I  |`  ( dom  R  u.  ran  R
) )  =  (  _I  |`  U. U. R
) )
2120sseq1d 3461 . . . . . . . . 9  |-  ( ( dom  R  u.  ran  R )  =  U. U. R  ->  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  ( t*rec `  R )  <->  (  _I  |` 
U. U. R )  C_  ( t*rec `  R ) ) )
2218, 21syl5ibr 221 . . . . . . . 8  |-  ( ( dom  R  u.  ran  R )  =  U. U. R  ->  ( ph  ->  (  _I  |`  ( dom  R  u.  ran  R ) )  C_  ( t*rec `  R )
) )
2317, 22mpcom 36 . . . . . . 7  |-  ( ph  ->  (  _I  |`  ( dom  R  u.  ran  R
) )  C_  (
t*rec `  R
) )
2413rtrclreclem2 12917 . . . . . . 7  |-  ( ph  ->  R  C_  ( t*rec `  R )
)
2514, 13rtrclreclem3 12918 . . . . . . 7  |-  ( ph  ->  ( ( t*rec
`  R )  o.  ( t*rec `  R ) )  C_  ( t*rec `  R ) )
26 fvex 5801 . . . . . . . 8  |-  ( t*rec `  R )  e.  _V
27 sseq2 3456 . . . . . . . . . . 11  |-  ( z  =  ( t*rec
`  R )  -> 
( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  <->  (  _I  |`  ( dom  R  u.  ran  R ) )  C_  ( t*rec `  R )
) )
28 sseq2 3456 . . . . . . . . . . 11  |-  ( z  =  ( t*rec
`  R )  -> 
( R  C_  z  <->  R 
C_  ( t*rec
`  R ) ) )
29 id 22 . . . . . . . . . . . . 13  |-  ( z  =  ( t*rec
`  R )  -> 
z  =  ( t*rec `  R )
)
3029, 29coeq12d 5097 . . . . . . . . . . . 12  |-  ( z  =  ( t*rec
`  R )  -> 
( z  o.  z
)  =  ( ( t*rec `  R
)  o.  ( t*rec `  R )
) )
3130, 29sseq12d 3463 . . . . . . . . . . 11  |-  ( z  =  ( t*rec
`  R )  -> 
( ( z  o.  z )  C_  z  <->  ( ( t*rec `  R )  o.  (
t*rec `  R
) )  C_  (
t*rec `  R
) ) )
3227, 28, 313anbi123d 1297 . . . . . . . . . 10  |-  ( z  =  ( t*rec
`  R )  -> 
( ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z )  <->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  (
t*rec `  R
)  /\  R  C_  (
t*rec `  R
)  /\  ( (
t*rec `  R
)  o.  ( t*rec `  R )
)  C_  ( t*rec `  R )
) ) )
3332a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( z  =  ( t*rec `  R
)  ->  ( (
(  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z )  <->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  (
t*rec `  R
)  /\  R  C_  (
t*rec `  R
)  /\  ( (
t*rec `  R
)  o.  ( t*rec `  R )
)  C_  ( t*rec `  R )
) ) ) )
3433alrimiv 1734 . . . . . . . 8  |-  ( ph  ->  A. z ( z  =  ( t*rec
`  R )  -> 
( ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z )  <->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  (
t*rec `  R
)  /\  R  C_  (
t*rec `  R
)  /\  ( (
t*rec `  R
)  o.  ( t*rec `  R )
)  C_  ( t*rec `  R )
) ) ) )
35 elabgt 3185 . . . . . . . 8  |-  ( ( ( t*rec `  R )  e.  _V  /\ 
A. z ( z  =  ( t*rec
`  R )  -> 
( ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z )  <->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  (
t*rec `  R
)  /\  R  C_  (
t*rec `  R
)  /\  ( (
t*rec `  R
)  o.  ( t*rec `  R )
)  C_  ( t*rec `  R )
) ) ) )  ->  ( ( t*rec `  R )  e.  { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } 
<->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  (
t*rec `  R
)  /\  R  C_  (
t*rec `  R
)  /\  ( (
t*rec `  R
)  o.  ( t*rec `  R )
)  C_  ( t*rec `  R )
) ) )
3626, 34, 35sylancr 661 . . . . . . 7  |-  ( ph  ->  ( ( t*rec
`  R )  e. 
{ z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } 
<->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  (
t*rec `  R
)  /\  R  C_  (
t*rec `  R
)  /\  ( (
t*rec `  R
)  o.  ( t*rec `  R )
)  C_  ( t*rec `  R )
) ) )
3723, 24, 25, 36mpbir3and 1177 . . . . . 6  |-  ( ph  ->  ( t*rec `  R )  e.  {
z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  (
z  o.  z ) 
C_  z ) } )
38 ne0i 3734 . . . . . 6  |-  ( ( t*rec `  R
)  e.  { z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  ->  { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  =/=  (/) )
3937, 38syl 16 . . . . 5  |-  ( ph  ->  { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  =/=  (/) )
40 intex 4538 . . . . 5  |-  ( { z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  (
z  o.  z ) 
C_  z ) }  =/=  (/)  <->  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  e.  _V )
4139, 40sylib 196 . . . 4  |-  ( ph  ->  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  e.  _V )
421, 12, 13, 41fvmptd 5879 . . 3  |-  ( ph  ->  ( ( x  e. 
_V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } ) `  R
)  =  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } )
43 intss1 4231 . . . . 5  |-  ( ( t*rec `  R
)  e.  { z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  ->  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  C_  ( t*rec
`  R ) )
4437, 43syl 16 . . . 4  |-  ( ph  ->  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  C_  ( t*rec
`  R ) )
45 vex 3054 . . . . . . . 8  |-  s  e. 
_V
46 sseq2 3456 . . . . . . . . 9  |-  ( z  =  s  ->  (
(  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  <->  (  _I  |`  ( dom  R  u.  ran  R ) )  C_  s )
)
47 sseq2 3456 . . . . . . . . 9  |-  ( z  =  s  ->  ( R  C_  z  <->  R  C_  s
) )
48 id 22 . . . . . . . . . . 11  |-  ( z  =  s  ->  z  =  s )
4948, 48coeq12d 5097 . . . . . . . . . 10  |-  ( z  =  s  ->  (
z  o.  z )  =  ( s  o.  s ) )
5049, 48sseq12d 3463 . . . . . . . . 9  |-  ( z  =  s  ->  (
( z  o.  z
)  C_  z  <->  ( s  o.  s )  C_  s
) )
5146, 47, 503anbi123d 1297 . . . . . . . 8  |-  ( z  =  s  ->  (
( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z )  <->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  s  /\  R  C_  s  /\  ( s  o.  s
)  C_  s )
) )
5245, 51elab 3188 . . . . . . 7  |-  ( s  e.  { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } 
<->  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  s  /\  R  C_  s  /\  ( s  o.  s
)  C_  s )
)
5314, 13rtrclreclem4 12919 . . . . . . . 8  |-  ( ph  ->  A. s ( ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  s  /\  R  C_  s  /\  ( s  o.  s
)  C_  s )  ->  ( t*rec `  R )  C_  s
) )
545319.21bi 1887 . . . . . . 7  |-  ( ph  ->  ( ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  s  /\  R  C_  s  /\  ( s  o.  s
)  C_  s )  ->  ( t*rec `  R )  C_  s
) )
5552, 54syl5bi 217 . . . . . 6  |-  ( ph  ->  ( s  e.  {
z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  (
z  o.  z ) 
C_  z ) }  ->  ( t*rec
`  R )  C_  s ) )
5655ralrimiv 2808 . . . . 5  |-  ( ph  ->  A. s  e.  {
z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  (
z  o.  z ) 
C_  z ) }  ( t*rec `  R )  C_  s
)
57 ssint 4232 . . . . 5  |-  ( ( t*rec `  R
)  C_  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } 
<-> 
A. s  e.  {
z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  (
z  o.  z ) 
C_  z ) }  ( t*rec `  R )  C_  s
)
5856, 57sylibr 212 . . . 4  |-  ( ph  ->  ( t*rec `  R )  C_  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R ) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) } )
5944, 58eqssd 3451 . . 3  |-  ( ph  ->  |^| { z  |  ( (  _I  |`  ( dom  R  u.  ran  R
) )  C_  z  /\  R  C_  z  /\  ( z  o.  z
)  C_  z ) }  =  ( t*rec `  R )
)
6042, 59eqtrd 2437 . 2  |-  ( ph  ->  ( ( x  e. 
_V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } ) `  R
)  =  ( t*rec `  R )
)
61 df-rtrcl 12849 . . 3  |-  t*  =  ( x  e.  _V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } )
62 fveq1 5790 . . . . 5  |-  ( t*  =  ( x  e.  _V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u. 
ran  x ) ) 
C_  z  /\  x  C_  z  /\  ( z  o.  z )  C_  z ) } )  ->  ( t* `  R )  =  ( ( x  e. 
_V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } ) `  R
) )
6362eqeq1d 2398 . . . 4  |-  ( t*  =  ( x  e.  _V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u. 
ran  x ) ) 
C_  z  /\  x  C_  z  /\  ( z  o.  z )  C_  z ) } )  ->  ( ( t* `  R )  =  ( t*rec
`  R )  <->  ( (
x  e.  _V  |->  |^|
{ z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } ) `  R
)  =  ( t*rec `  R )
) )
6463imbi2d 314 . . 3  |-  ( t*  =  ( x  e.  _V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u. 
ran  x ) ) 
C_  z  /\  x  C_  z  /\  ( z  o.  z )  C_  z ) } )  ->  ( ( ph  ->  ( t* `  R )  =  ( t*rec `  R
) )  <->  ( ph  ->  ( ( x  e. 
_V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } ) `  R
)  =  ( t*rec `  R )
) ) )
6561, 64ax-mp 5 . 2  |-  ( (
ph  ->  ( t* `  R )  =  ( t*rec `  R ) )  <->  ( ph  ->  ( ( x  e. 
_V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u.  ran  x
) )  C_  z  /\  x  C_  z  /\  ( z  o.  z
)  C_  z ) } ) `  R
)  =  ( t*rec `  R )
) )
6660, 65mpbir 209 1  |-  ( ph  ->  ( t* `  R )  =  ( t*rec `  R
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971   A.wal 1397    = wceq 1399    e. wcel 1836   {cab 2381    =/= wne 2591   A.wral 2746   _Vcvv 3051    u. cun 3404    C_ wss 3406   (/)c0 3728   U.cuni 4180   |^|cint 4216    |-> cmpt 4442    _I cid 4721   dom cdm 4930   ran crn 4931    |` cres 4932    o. ccom 4934   Rel wrel 4935   ` cfv 5513   t*crtcl 12847   t*reccrtrcl 12913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-2nd 6722  df-recs 6982  df-rdg 7016  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-nn 10475  df-2 10533  df-n0 10735  df-z 10804  df-uz 11024  df-seq 12034  df-rtrcl 12849  df-relexp 12881  df-rtrclrec 12914
This theorem is referenced by:  rtrclind  12923
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