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Theorem usgedgvadf1 32772
Description: The mapping of edges containing a given vertex into the set of vertices is 1-1, analogous to usgraidx2v 24539. The edge is mapped to the other vertex of the edge containing the vertex N. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
Hypotheses
Ref Expression
usgedgvadf1.v  |-  V  =  ( Vtx  `  G )
usgedgvadf1.e  |-  E  =  ( Edges  `  G )
usgedgvadf1.a  |-  A  =  { e  e.  E  |  N  e.  e }
usgedgvadf1.f  |-  F  =  ( e  e.  A  |->  ( iota_ m  e.  V  e  =  { m ,  N } ) )
Assertion
Ref Expression
usgedgvadf1  |-  ( ( G  e. USGrph  /\  N  e.  V )  ->  F : A -1-1-> V )
Distinct variable groups:    e, E, m    m, G    e, N, m    m, V    A, e    e, G, m    e, V
Allowed substitution hints:    A( m)    F( e, m)

Proof of Theorem usgedgvadf1
Dummy variables  b 
c  w  z  y  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgedgvadf1.v . . . . 5  |-  V  =  ( Vtx  `  G )
2 usgedgvadf1.e . . . . 5  |-  E  =  ( Edges  `  G )
3 usgedgvadf1.a . . . . 5  |-  A  =  { e  e.  E  |  N  e.  e }
41, 2, 3usgedgvadf1lem1 32770 . . . 4  |-  ( ( G  e. USGrph  /\  b  e.  A )  ->  ( iota_ m  e.  V  b  =  { m ,  N } )  e.  V )
54ralrimiva 2810 . . 3  |-  ( G  e. USGrph  ->  A. b  e.  A  ( iota_ m  e.  V  b  =  { m ,  N } )  e.  V )
65adantr 463 . 2  |-  ( ( G  e. USGrph  /\  N  e.  V )  ->  A. b  e.  A  ( iota_ m  e.  V  b  =  { m ,  N } )  e.  V
)
7 simpl 455 . . . . . . . . 9  |-  ( ( G  e. USGrph  /\  N  e.  V )  ->  G  e. USGrph  )
8 simpl 455 . . . . . . . . 9  |-  ( ( b  e.  A  /\  c  e.  A )  ->  b  e.  A )
97, 8anim12i 564 . . . . . . . 8  |-  ( ( ( G  e. USGrph  /\  N  e.  V )  /\  (
b  e.  A  /\  c  e.  A )
)  ->  ( G  e. USGrph  /\  b  e.  A
) )
10 preq1 4040 . . . . . . . . . 10  |-  ( x  =  z  ->  { x ,  N }  =  {
z ,  N }
)
1110eqeq2d 2410 . . . . . . . . 9  |-  ( x  =  z  ->  (
b  =  { x ,  N }  <->  b  =  { z ,  N } ) )
1211cbvriotav 6191 . . . . . . . 8  |-  ( iota_ x  e.  V  b  =  { x ,  N } )  =  (
iota_ z  e.  V  b  =  { z ,  N } )
131, 2, 3usgedgvadf1lem2 32771 . . . . . . . 8  |-  ( ( G  e. USGrph  /\  b  e.  A )  ->  (
( iota_ x  e.  V  b  =  { x ,  N } )  =  ( iota_ z  e.  V  b  =  { z ,  N } )  -> 
b  =  { (
iota_ x  e.  V  b  =  { x ,  N } ) ,  N } ) )
149, 12, 13mpisyl 18 . . . . . . 7  |-  ( ( ( G  e. USGrph  /\  N  e.  V )  /\  (
b  e.  A  /\  c  e.  A )
)  ->  b  =  { ( iota_ x  e.  V  b  =  {
x ,  N }
) ,  N }
)
15 simpr 459 . . . . . . . . 9  |-  ( ( b  e.  A  /\  c  e.  A )  ->  c  e.  A )
167, 15anim12i 564 . . . . . . . 8  |-  ( ( ( G  e. USGrph  /\  N  e.  V )  /\  (
b  e.  A  /\  c  e.  A )
)  ->  ( G  e. USGrph  /\  c  e.  A
) )
17 preq1 4040 . . . . . . . . . 10  |-  ( y  =  w  ->  { y ,  N }  =  { w ,  N } )
1817eqeq2d 2410 . . . . . . . . 9  |-  ( y  =  w  ->  (
c  =  { y ,  N }  <->  c  =  { w ,  N } ) )
1918cbvriotav 6191 . . . . . . . 8  |-  ( iota_ y  e.  V  c  =  { y ,  N } )  =  (
iota_ w  e.  V  c  =  { w ,  N } )
201, 2, 3usgedgvadf1lem2 32771 . . . . . . . 8  |-  ( ( G  e. USGrph  /\  c  e.  A )  ->  (
( iota_ y  e.  V  c  =  { y ,  N } )  =  ( iota_ w  e.  V  c  =  { w ,  N } )  -> 
c  =  { (
iota_ y  e.  V  c  =  { y ,  N } ) ,  N } ) )
2116, 19, 20mpisyl 18 . . . . . . 7  |-  ( ( ( G  e. USGrph  /\  N  e.  V )  /\  (
b  e.  A  /\  c  e.  A )
)  ->  c  =  { ( iota_ y  e.  V  c  =  {
y ,  N }
) ,  N }
)
2214, 21eqeq12d 2418 . . . . . 6  |-  ( ( ( G  e. USGrph  /\  N  e.  V )  /\  (
b  e.  A  /\  c  e.  A )
)  ->  ( b  =  c  <->  { ( iota_ x  e.  V  b  =  {
x ,  N }
) ,  N }  =  { ( iota_ y  e.  V  c  =  {
y ,  N }
) ,  N }
) )
2322notbid 292 . . . . 5  |-  ( ( ( G  e. USGrph  /\  N  e.  V )  /\  (
b  e.  A  /\  c  e.  A )
)  ->  ( -.  b  =  c  <->  -.  { (
iota_ x  e.  V  b  =  { x ,  N } ) ,  N }  =  {
( iota_ y  e.  V  c  =  { y ,  N } ) ,  N } ) )
24 riotaex 6184 . . . . . . . . . . 11  |-  ( iota_ x  e.  V  b  =  { x ,  N } )  e.  _V
2524a1i 11 . . . . . . . . . 10  |-  ( N  e.  V  ->  ( iota_ x  e.  V  b  =  { x ,  N } )  e. 
_V )
26 id 22 . . . . . . . . . 10  |-  ( N  e.  V  ->  N  e.  V )
27 riotaex 6184 . . . . . . . . . . 11  |-  ( iota_ y  e.  V  c  =  { y ,  N } )  e.  _V
2827a1i 11 . . . . . . . . . 10  |-  ( N  e.  V  ->  ( iota_ y  e.  V  c  =  { y ,  N } )  e. 
_V )
29 preq12bg 4140 . . . . . . . . . 10  |-  ( ( ( ( iota_ x  e.  V  b  =  {
x ,  N }
)  e.  _V  /\  N  e.  V )  /\  ( ( iota_ y  e.  V  c  =  {
y ,  N }
)  e.  _V  /\  N  e.  V )
)  ->  ( {
( iota_ x  e.  V  b  =  { x ,  N } ) ,  N }  =  {
( iota_ y  e.  V  c  =  { y ,  N } ) ,  N }  <->  ( (
( iota_ x  e.  V  b  =  { x ,  N } )  =  ( iota_ y  e.  V  c  =  { y ,  N } )  /\  N  =  N )  \/  ( ( iota_ x  e.  V  b  =  {
x ,  N }
)  =  N  /\  N  =  ( iota_ y  e.  V  c  =  { y ,  N } ) ) ) ) )
3025, 26, 28, 26, 29syl22anc 1227 . . . . . . . . 9  |-  ( N  e.  V  ->  ( { ( iota_ x  e.  V  b  =  {
x ,  N }
) ,  N }  =  { ( iota_ y  e.  V  c  =  {
y ,  N }
) ,  N }  <->  ( ( ( iota_ x  e.  V  b  =  {
x ,  N }
)  =  ( iota_ y  e.  V  c  =  { y ,  N } )  /\  N  =  N )  \/  (
( iota_ x  e.  V  b  =  { x ,  N } )  =  N  /\  N  =  ( iota_ y  e.  V  c  =  { y ,  N } ) ) ) ) )
3130notbid 292 . . . . . . . 8  |-  ( N  e.  V  ->  ( -.  { ( iota_ x  e.  V  b  =  {
x ,  N }
) ,  N }  =  { ( iota_ y  e.  V  c  =  {
y ,  N }
) ,  N }  <->  -.  ( ( ( iota_ x  e.  V  b  =  { x ,  N } )  =  (
iota_ y  e.  V  c  =  { y ,  N } )  /\  N  =  N )  \/  ( ( iota_ x  e.  V  b  =  {
x ,  N }
)  =  N  /\  N  =  ( iota_ y  e.  V  c  =  { y ,  N } ) ) ) ) )
3231adantl 464 . . . . . . 7  |-  ( ( G  e. USGrph  /\  N  e.  V )  ->  ( -.  { ( iota_ x  e.  V  b  =  {
x ,  N }
) ,  N }  =  { ( iota_ y  e.  V  c  =  {
y ,  N }
) ,  N }  <->  -.  ( ( ( iota_ x  e.  V  b  =  { x ,  N } )  =  (
iota_ y  e.  V  c  =  { y ,  N } )  /\  N  =  N )  \/  ( ( iota_ x  e.  V  b  =  {
x ,  N }
)  =  N  /\  N  =  ( iota_ y  e.  V  c  =  { y ,  N } ) ) ) ) )
33 ioran 488 . . . . . . . . . 10  |-  ( -.  ( ( ( iota_ x  e.  V  b  =  { x ,  N } )  =  (
iota_ y  e.  V  c  =  { y ,  N } )  /\  N  =  N )  \/  ( ( iota_ x  e.  V  b  =  {
x ,  N }
)  =  N  /\  N  =  ( iota_ y  e.  V  c  =  { y ,  N } ) ) )  <-> 
( -.  ( (
iota_ x  e.  V  b  =  { x ,  N } )  =  ( iota_ y  e.  V  c  =  { y ,  N } )  /\  N  =  N )  /\  -.  ( ( iota_ x  e.  V  b  =  { x ,  N } )  =  N  /\  N  =  (
iota_ y  e.  V  c  =  { y ,  N } ) ) ) )
34 ianor 486 . . . . . . . . . . . 12  |-  ( -.  ( ( iota_ x  e.  V  b  =  {
x ,  N }
)  =  ( iota_ y  e.  V  c  =  { y ,  N } )  /\  N  =  N )  <->  ( -.  ( iota_ x  e.  V  b  =  { x ,  N } )  =  ( iota_ y  e.  V  c  =  { y ,  N } )  \/ 
-.  N  =  N ) )
35 preq1 4040 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  m  ->  { x ,  N }  =  {
m ,  N }
)
3635eqeq2d 2410 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  m  ->  (
b  =  { x ,  N }  <->  b  =  { m ,  N } ) )
3736cbvriotav 6191 . . . . . . . . . . . . . . . . 17  |-  ( iota_ x  e.  V  b  =  { x ,  N } )  =  (
iota_ m  e.  V  b  =  { m ,  N } )
38 preq1 4040 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  m  ->  { y ,  N }  =  { m ,  N } )
3938eqeq2d 2410 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  m  ->  (
c  =  { y ,  N }  <->  c  =  { m ,  N } ) )
4039cbvriotav 6191 . . . . . . . . . . . . . . . . 17  |-  ( iota_ y  e.  V  c  =  { y ,  N } )  =  (
iota_ m  e.  V  c  =  { m ,  N } )
4137, 40eqeq12i 2416 . . . . . . . . . . . . . . . 16  |-  ( (
iota_ x  e.  V  b  =  { x ,  N } )  =  ( iota_ y  e.  V  c  =  { y ,  N } )  <->  ( iota_ m  e.  V  b  =  { m ,  N } )  =  (
iota_ m  e.  V  c  =  { m ,  N } ) )
4241notbii 294 . . . . . . . . . . . . . . 15  |-  ( -.  ( iota_ x  e.  V  b  =  { x ,  N } )  =  ( iota_ y  e.  V  c  =  { y ,  N } )  <->  -.  ( iota_ m  e.  V  b  =  { m ,  N } )  =  ( iota_ m  e.  V  c  =  { m ,  N } ) )
4342biimpi 194 . . . . . . . . . . . . . 14  |-  ( -.  ( iota_ x  e.  V  b  =  { x ,  N } )  =  ( iota_ y  e.  V  c  =  { y ,  N } )  ->  -.  ( iota_ m  e.  V  b  =  { m ,  N } )  =  ( iota_ m  e.  V  c  =  { m ,  N } ) )
4443a1d 25 . . . . . . . . . . . . 13  |-  ( -.  ( iota_ x  e.  V  b  =  { x ,  N } )  =  ( iota_ y  e.  V  c  =  { y ,  N } )  -> 
( G  e. USGrph  ->  -.  ( iota_ m  e.  V  b  =  { m ,  N } )  =  ( iota_ m  e.  V  c  =  { m ,  N } ) ) )
45 eqid 2396 . . . . . . . . . . . . . 14  |-  N  =  N
4645pm2.24i 144 . . . . . . . . . . . . 13  |-  ( -.  N  =  N  -> 
( G  e. USGrph  ->  -.  ( iota_ m  e.  V  b  =  { m ,  N } )  =  ( iota_ m  e.  V  c  =  { m ,  N } ) ) )
4744, 46jaoi 377 . . . . . . . . . . . 12  |-  ( ( -.  ( iota_ x  e.  V  b  =  {
x ,  N }
)  =  ( iota_ y  e.  V  c  =  { y ,  N } )  \/  -.  N  =  N )  ->  ( G  e. USGrph  ->  -.  ( iota_ m  e.  V  b  =  { m ,  N } )  =  ( iota_ m  e.  V  c  =  { m ,  N } ) ) )
4834, 47sylbi 195 . . . . . . . . . . 11  |-  ( -.  ( ( iota_ x  e.  V  b  =  {
x ,  N }
)  =  ( iota_ y  e.  V  c  =  { y ,  N } )  /\  N  =  N )  ->  ( G  e. USGrph  ->  -.  ( iota_ m  e.  V  b  =  { m ,  N } )  =  ( iota_ m  e.  V  c  =  { m ,  N } ) ) )
4948adantr 463 . . . . . . . . . 10  |-  ( ( -.  ( ( iota_ x  e.  V  b  =  { x ,  N } )  =  (
iota_ y  e.  V  c  =  { y ,  N } )  /\  N  =  N )  /\  -.  ( ( iota_ x  e.  V  b  =  { x ,  N } )  =  N  /\  N  =  (
iota_ y  e.  V  c  =  { y ,  N } ) ) )  ->  ( G  e. USGrph  ->  -.  ( iota_ m  e.  V  b  =  { m ,  N } )  =  (
iota_ m  e.  V  c  =  { m ,  N } ) ) )
5033, 49sylbi 195 . . . . . . . . 9  |-  ( -.  ( ( ( iota_ x  e.  V  b  =  { x ,  N } )  =  (
iota_ y  e.  V  c  =  { y ,  N } )  /\  N  =  N )  \/  ( ( iota_ x  e.  V  b  =  {
x ,  N }
)  =  N  /\  N  =  ( iota_ y  e.  V  c  =  { y ,  N } ) ) )  ->  ( G  e. USGrph  ->  -.  ( iota_ m  e.  V  b  =  {
m ,  N }
)  =  ( iota_ m  e.  V  c  =  { m ,  N } ) ) )
5150com12 31 . . . . . . . 8  |-  ( G  e. USGrph  ->  ( -.  (
( ( iota_ x  e.  V  b  =  {
x ,  N }
)  =  ( iota_ y  e.  V  c  =  { y ,  N } )  /\  N  =  N )  \/  (
( iota_ x  e.  V  b  =  { x ,  N } )  =  N  /\  N  =  ( iota_ y  e.  V  c  =  { y ,  N } ) ) )  ->  -.  ( iota_ m  e.  V  b  =  { m ,  N } )  =  ( iota_ m  e.  V  c  =  { m ,  N } ) ) )
5251adantr 463 . . . . . . 7  |-  ( ( G  e. USGrph  /\  N  e.  V )  ->  ( -.  ( ( ( iota_ x  e.  V  b  =  { x ,  N } )  =  (
iota_ y  e.  V  c  =  { y ,  N } )  /\  N  =  N )  \/  ( ( iota_ x  e.  V  b  =  {
x ,  N }
)  =  N  /\  N  =  ( iota_ y  e.  V  c  =  { y ,  N } ) ) )  ->  -.  ( iota_ m  e.  V  b  =  { m ,  N } )  =  (
iota_ m  e.  V  c  =  { m ,  N } ) ) )
5332, 52sylbid 215 . . . . . 6  |-  ( ( G  e. USGrph  /\  N  e.  V )  ->  ( -.  { ( iota_ x  e.  V  b  =  {
x ,  N }
) ,  N }  =  { ( iota_ y  e.  V  c  =  {
y ,  N }
) ,  N }  ->  -.  ( iota_ m  e.  V  b  =  {
m ,  N }
)  =  ( iota_ m  e.  V  c  =  { m ,  N } ) ) )
5453adantr 463 . . . . 5  |-  ( ( ( G  e. USGrph  /\  N  e.  V )  /\  (
b  e.  A  /\  c  e.  A )
)  ->  ( -.  { ( iota_ x  e.  V  b  =  { x ,  N } ) ,  N }  =  {
( iota_ y  e.  V  c  =  { y ,  N } ) ,  N }  ->  -.  ( iota_ m  e.  V  b  =  { m ,  N } )  =  ( iota_ m  e.  V  c  =  { m ,  N } ) ) )
5523, 54sylbid 215 . . . 4  |-  ( ( ( G  e. USGrph  /\  N  e.  V )  /\  (
b  e.  A  /\  c  e.  A )
)  ->  ( -.  b  =  c  ->  -.  ( iota_ m  e.  V  b  =  { m ,  N } )  =  ( iota_ m  e.  V  c  =  { m ,  N } ) ) )
5655con4d 105 . . 3  |-  ( ( ( G  e. USGrph  /\  N  e.  V )  /\  (
b  e.  A  /\  c  e.  A )
)  ->  ( ( iota_ m  e.  V  b  =  { m ,  N } )  =  ( iota_ m  e.  V  c  =  { m ,  N } )  -> 
b  =  c ) )
5756ralrimivva 2817 . 2  |-  ( ( G  e. USGrph  /\  N  e.  V )  ->  A. b  e.  A  A. c  e.  A  ( ( iota_ m  e.  V  b  =  { m ,  N } )  =  ( iota_ m  e.  V  c  =  { m ,  N } )  -> 
b  =  c ) )
58 usgedgvadf1.f . . . 4  |-  F  =  ( e  e.  A  |->  ( iota_ m  e.  V  e  =  { m ,  N } ) )
59 simpl 455 . . . . . . 7  |-  ( ( e  =  b  /\  m  e.  V )  ->  e  =  b )
6059eqeq1d 2398 . . . . . 6  |-  ( ( e  =  b  /\  m  e.  V )  ->  ( e  =  {
m ,  N }  <->  b  =  { m ,  N } ) )
6160riotabidva 6196 . . . . 5  |-  ( e  =  b  ->  ( iota_ m  e.  V  e  =  { m ,  N } )  =  ( iota_ m  e.  V  b  =  { m ,  N } ) )
6261cbvmptv 4475 . . . 4  |-  ( e  e.  A  |->  ( iota_ m  e.  V  e  =  { m ,  N } ) )  =  ( b  e.  A  |->  ( iota_ m  e.  V  b  =  { m ,  N } ) )
6358, 62eqtri 2425 . . 3  |-  F  =  ( b  e.  A  |->  ( iota_ m  e.  V  b  =  { m ,  N } ) )
64 eqeq1 2400 . . . 4  |-  ( b  =  c  ->  (
b  =  { m ,  N }  <->  c  =  { m ,  N } ) )
6564riotabidv 6182 . . 3  |-  ( b  =  c  ->  ( iota_ m  e.  V  b  =  { m ,  N } )  =  ( iota_ m  e.  V  c  =  { m ,  N } ) )
6663, 65f1mpt 6090 . 2  |-  ( F : A -1-1-> V  <->  ( A. b  e.  A  ( iota_ m  e.  V  b  =  { m ,  N } )  e.  V  /\  A. b  e.  A  A. c  e.  A  ( ( iota_ m  e.  V  b  =  { m ,  N } )  =  ( iota_ m  e.  V  c  =  { m ,  N } )  -> 
b  =  c ) ) )
676, 57, 66sylanbrc 662 1  |-  ( ( G  e. USGrph  /\  N  e.  V )  ->  F : A -1-1-> V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1399    e. wcel 1836   A.wral 2746   {crab 2750   _Vcvv 3051   {cpr 3963    |-> cmpt 4442   -1-1->wf1 5510   ` cfv 5513   iota_crio 6179   USGrph cusg 24476   Edges cedg 24477   Vtx cvtx 32738
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-rmo 2754  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-1st 6721  df-2nd 6722  df-recs 6982  df-rdg 7016  df-1o 7070  df-2o 7071  df-oadd 7074  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-fin 7461  df-card 8255  df-cda 8483  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-fz 11616  df-hash 12331  df-usgra 24479  df-edg 24482  df-vtx 32739
This theorem is referenced by:  usgedgleord  32776
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