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Theorem usgedgvadf1ALT 32790
Description: The mapping of edges containing a given vertex into the set of vertices is 1-1, analogous to usgraidx2v 24595. 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.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
usgedgvadf1ALT.v  |-  V  =  ( 1st `  G
)
usgedgvadf1ALT.e  |-  E  =  ( Edges  `  G )
usgedgvadf1ALT.a  |-  A  =  { e  e.  E  |  N  e.  e }
usgedgvadf1ALT.f  |-  F  =  ( e  e.  A  |->  ( iota_ m  e.  V  e  =  { m ,  N } ) )
Assertion
Ref Expression
usgedgvadf1ALT  |-  ( ( 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 usgedgvadf1ALT
Dummy variables  b 
c  w  z  y  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgedgvadf1ALT.v . . . . 5  |-  V  =  ( 1st `  G
)
2 usgedgvadf1ALT.e . . . . 5  |-  E  =  ( Edges  `  G )
3 usgedgvadf1ALT.a . . . . 5  |-  A  =  { e  e.  E  |  N  e.  e }
41, 2, 3usgedgvadf1ALTlem1 32788 . . . 4  |-  ( ( G  e. USGrph  /\  b  e.  A )  ->  ( iota_ m  e.  V  b  =  { m ,  N } )  e.  V )
54ralrimiva 2868 . . 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 4095 . . . . . . . . . 10  |-  ( x  =  z  ->  { x ,  N }  =  {
z ,  N }
)
1110eqeq2d 2468 . . . . . . . . 9  |-  ( x  =  z  ->  (
b  =  { x ,  N }  <->  b  =  { z ,  N } ) )
1211cbvriotav 6243 . . . . . . . 8  |-  ( iota_ x  e.  V  b  =  { x ,  N } )  =  (
iota_ z  e.  V  b  =  { z ,  N } )
131, 2, 3usgedgvadf1ALTlem2 32789 . . . . . . . 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 4095 . . . . . . . . . 10  |-  ( y  =  w  ->  { y ,  N }  =  { w ,  N } )
1817eqeq2d 2468 . . . . . . . . 9  |-  ( y  =  w  ->  (
c  =  { y ,  N }  <->  c  =  { w ,  N } ) )
1918cbvriotav 6243 . . . . . . . 8  |-  ( iota_ y  e.  V  c  =  { y ,  N } )  =  (
iota_ w  e.  V  c  =  { w ,  N } )
201, 2, 3usgedgvadf1ALTlem2 32789 . . . . . . . 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 2476 . . . . . 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 6236 . . . . . . . . . . 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 6236 . . . . . . . . . . 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 4195 . . . . . . . . . 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 4095 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  m  ->  { x ,  N }  =  {
m ,  N }
)
3635eqeq2d 2468 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  m  ->  (
b  =  { x ,  N }  <->  b  =  { m ,  N } ) )
3736cbvriotav 6243 . . . . . . . . . . . . . . . . 17  |-  ( iota_ x  e.  V  b  =  { x ,  N } )  =  (
iota_ m  e.  V  b  =  { m ,  N } )
38 preq1 4095 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  m  ->  { y ,  N }  =  { m ,  N } )
3938eqeq2d 2468 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  m  ->  (
c  =  { y ,  N }  <->  c  =  { m ,  N } ) )
4039cbvriotav 6243 . . . . . . . . . . . . . . . . 17  |-  ( iota_ y  e.  V  c  =  { y ,  N } )  =  (
iota_ m  e.  V  c  =  { m ,  N } )
4137, 40eqeq12i 2474 . . . . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . 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 2875 . 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 usgedgvadf1ALT.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 2456 . . . . . 6  |-  ( ( e  =  b  /\  m  e.  V )  ->  ( e  =  {
m ,  N }  <->  b  =  { m ,  N } ) )
6160riotabidva 6248 . . . . 5  |-  ( e  =  b  ->  ( iota_ m  e.  V  e  =  { m ,  N } )  =  ( iota_ m  e.  V  b  =  { m ,  N } ) )
6261cbvmptv 4530 . . . 4  |-  ( e  e.  A  |->  ( iota_ m  e.  V  e  =  { m ,  N } ) )  =  ( b  e.  A  |->  ( iota_ m  e.  V  b  =  { m ,  N } ) )
6358, 62eqtri 2483 . . 3  |-  F  =  ( b  e.  A  |->  ( iota_ m  e.  V  b  =  { m ,  N } ) )
64 eqeq1 2458 . . . 4  |-  ( b  =  c  ->  (
b  =  { m ,  N }  <->  c  =  { m ,  N } ) )
6564riotabidv 6234 . . 3  |-  ( b  =  c  ->  ( iota_ m  e.  V  b  =  { m ,  N } )  =  ( iota_ m  e.  V  c  =  { m ,  N } ) )
6663, 65f1mpt 6144 . 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 1398    e. wcel 1823   A.wral 2804   {crab 2808   _Vcvv 3106   {cpr 4018    |-> cmpt 4497   -1-1->wf1 5567   ` cfv 5570   iota_crio 6231   1stc1st 6771   USGrph cusg 24532   Edges cedg 24533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12388  df-usgra 24535  df-edg 24538
This theorem is referenced by:  usgedgleordALT  32792
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