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Theorem usgreg2spot 24772
Description: In a finite k-regular graph the set of all paths of length two is the union of all the paths of length 2 over the vertices which are in the middle of such a path. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
Hypothesis
Ref Expression
usgreghash2spot.m  |-  M  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) } )
Assertion
Ref Expression
usgreg2spot  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  ( A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K  ->  ( V 2SPathOnOt  E )  =  U_ x  e.  V  ( M `  x )
) )
Distinct variable groups:    t, E, x, a    V, a, t, x    E, a, v, t   
v, V, a    x, K    v, M    x, v
Allowed substitution hints:    K( v, t, a)    M( x, t, a)

Proof of Theorem usgreg2spot
Dummy variables  p  y  z  f  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrav 24042 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 el2pthsot 24585 . . . . . . . . . 10  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( p  e.  ( V 2SPathOnOt  E )  <->  E. y  e.  V  E. x  e.  V  E. z  e.  V  ( p  =  <. y ,  x ,  z >.  /\  E. f E. q ( f ( V SPaths  E ) q  /\  ( # `  f )  =  2  /\  ( y  =  ( q `  0
)  /\  x  =  ( q `  1
)  /\  z  =  ( q `  2
) ) ) ) ) )
31, 2syl 16 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( p  e.  ( V 2SPathOnOt  E )  <->  E. y  e.  V  E. x  e.  V  E. z  e.  V  (
p  =  <. y ,  x ,  z >.  /\  E. f E. q
( f ( V SPaths  E ) q  /\  ( # `  f )  =  2  /\  (
y  =  ( q `
 0 )  /\  x  =  ( q `  1 )  /\  z  =  ( q `  2 ) ) ) ) ) )
4 simpr 461 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  y  e.  V )  ->  y  e.  V )
54ad2antrr 725 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  y  e.  V )
6 simpr 461 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( V USGrph  E  /\  y  e.  V )  /\  x  e.  V
)  ->  x  e.  V )
76adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  x  e.  V )
8 simpr 461 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  z  e.  V )
95, 7, 83jca 1176 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  (
y  e.  V  /\  x  e.  V  /\  z  e.  V )
)
10 ot2ndg 6799 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  V  /\  x  e.  V  /\  z  e.  V )  ->  ( 2nd `  ( 1st `  <. y ,  x ,  z >. )
)  =  x )
119, 10syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  ( 2nd `  ( 1st `  <. y ,  x ,  z
>. ) )  =  x )
12 eqidd 2468 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  <. y ,  x ,  z >.  =  <. y ,  x ,  z >. )
13 otel3xp 5035 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. y ,  x ,  z >.  =  <. y ,  x ,  z
>.  /\  ( y  e.  V  /\  x  e.  V  /\  z  e.  V ) )  ->  <. y ,  x ,  z >.  e.  (
( V  X.  V
)  X.  V ) )
1412, 9, 13syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  <. y ,  x ,  z >.  e.  ( ( V  X.  V )  X.  V
) )
1511, 14jca 532 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  (
( 2nd `  ( 1st `  <. y ,  x ,  z >. )
)  =  x  /\  <.
y ,  x ,  z >.  e.  (
( V  X.  V
)  X.  V ) ) )
16 fveq2 5866 . . . . . . . . . . . . . . . . . . 19  |-  ( p  =  <. y ,  x ,  z >.  ->  ( 1st `  p )  =  ( 1st `  <. y ,  x ,  z
>. ) )
1716fveq2d 5870 . . . . . . . . . . . . . . . . . 18  |-  ( p  =  <. y ,  x ,  z >.  ->  ( 2nd `  ( 1st `  p
) )  =  ( 2nd `  ( 1st `  <. y ,  x ,  z >. )
) )
1817eqeq1d 2469 . . . . . . . . . . . . . . . . 17  |-  ( p  =  <. y ,  x ,  z >.  ->  (
( 2nd `  ( 1st `  p ) )  =  x  <->  ( 2nd `  ( 1st `  <. y ,  x ,  z
>. ) )  =  x ) )
19 eleq1 2539 . . . . . . . . . . . . . . . . 17  |-  ( p  =  <. y ,  x ,  z >.  ->  (
p  e.  ( ( V  X.  V )  X.  V )  <->  <. y ,  x ,  z >.  e.  ( ( V  X.  V )  X.  V
) ) )
2018, 19anbi12d 710 . . . . . . . . . . . . . . . 16  |-  ( p  =  <. y ,  x ,  z >.  ->  (
( ( 2nd `  ( 1st `  p ) )  =  x  /\  p  e.  ( ( V  X.  V )  X.  V
) )  <->  ( ( 2nd `  ( 1st `  <. y ,  x ,  z
>. ) )  =  x  /\  <. y ,  x ,  z >.  e.  ( ( V  X.  V
)  X.  V ) ) ) )
2115, 20syl5ibr 221 . . . . . . . . . . . . . . 15  |-  ( p  =  <. y ,  x ,  z >.  ->  (
( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  (
( 2nd `  ( 1st `  p ) )  =  x  /\  p  e.  ( ( V  X.  V )  X.  V
) ) ) )
2221adantr 465 . . . . . . . . . . . . . 14  |-  ( ( p  =  <. y ,  x ,  z >.  /\  E. f E. q
( f ( V SPaths  E ) q  /\  ( # `  f )  =  2  /\  (
y  =  ( q `
 0 )  /\  x  =  ( q `  1 )  /\  z  =  ( q `  2 ) ) ) )  ->  (
( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  (
( 2nd `  ( 1st `  p ) )  =  x  /\  p  e.  ( ( V  X.  V )  X.  V
) ) ) )
2322com12 31 . . . . . . . . . . . . 13  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  (
( p  =  <. y ,  x ,  z
>.  /\  E. f E. q ( f ( V SPaths  E ) q  /\  ( # `  f
)  =  2  /\  ( y  =  ( q `  0 )  /\  x  =  ( q `  1 )  /\  z  =  ( q `  2 ) ) ) )  -> 
( ( 2nd `  ( 1st `  p ) )  =  x  /\  p  e.  ( ( V  X.  V )  X.  V
) ) ) )
2423rexlimdva 2955 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  E  /\  y  e.  V )  /\  x  e.  V
)  ->  ( E. z  e.  V  (
p  =  <. y ,  x ,  z >.  /\  E. f E. q
( f ( V SPaths  E ) q  /\  ( # `  f )  =  2  /\  (
y  =  ( q `
 0 )  /\  x  =  ( q `  1 )  /\  z  =  ( q `  2 ) ) ) )  ->  (
( 2nd `  ( 1st `  p ) )  =  x  /\  p  e.  ( ( V  X.  V )  X.  V
) ) ) )
2524reximdva 2938 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  y  e.  V )  ->  ( E. x  e.  V  E. z  e.  V  ( p  =  <. y ,  x ,  z
>.  /\  E. f E. q ( f ( V SPaths  E ) q  /\  ( # `  f
)  =  2  /\  ( y  =  ( q `  0 )  /\  x  =  ( q `  1 )  /\  z  =  ( q `  2 ) ) ) )  ->  E. x  e.  V  ( ( 2nd `  ( 1st `  p ) )  =  x  /\  p  e.  ( ( V  X.  V )  X.  V
) ) ) )
26 r19.41v 3014 . . . . . . . . . . 11  |-  ( E. x  e.  V  ( ( 2nd `  ( 1st `  p ) )  =  x  /\  p  e.  ( ( V  X.  V )  X.  V
) )  <->  ( E. x  e.  V  ( 2nd `  ( 1st `  p
) )  =  x  /\  p  e.  ( ( V  X.  V
)  X.  V ) ) )
2725, 26syl6ib 226 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  y  e.  V )  ->  ( E. x  e.  V  E. z  e.  V  ( p  =  <. y ,  x ,  z
>.  /\  E. f E. q ( f ( V SPaths  E ) q  /\  ( # `  f
)  =  2  /\  ( y  =  ( q `  0 )  /\  x  =  ( q `  1 )  /\  z  =  ( q `  2 ) ) ) )  -> 
( E. x  e.  V  ( 2nd `  ( 1st `  p ) )  =  x  /\  p  e.  ( ( V  X.  V )  X.  V
) ) ) )
2827rexlimdva 2955 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( E. y  e.  V  E. x  e.  V  E. z  e.  V  ( p  =  <. y ,  x ,  z >.  /\  E. f E. q ( f ( V SPaths  E ) q  /\  ( # `  f )  =  2  /\  ( y  =  ( q `  0
)  /\  x  =  ( q `  1
)  /\  z  =  ( q `  2
) ) ) )  ->  ( E. x  e.  V  ( 2nd `  ( 1st `  p
) )  =  x  /\  p  e.  ( ( V  X.  V
)  X.  V ) ) ) )
293, 28sylbid 215 . . . . . . . 8  |-  ( V USGrph  E  ->  ( p  e.  ( V 2SPathOnOt  E )  ->  ( E. x  e.  V  ( 2nd `  ( 1st `  p ) )  =  x  /\  p  e.  ( ( V  X.  V )  X.  V
) ) ) )
3029ad2antrr 725 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  ->  (
p  e.  ( V 2SPathOnOt  E )  ->  ( E. x  e.  V  ( 2nd `  ( 1st `  p ) )  =  x  /\  p  e.  ( ( V  X.  V )  X.  V
) ) ) )
3130pm4.71rd 635 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  ->  (
p  e.  ( V 2SPathOnOt  E )  <->  ( ( E. x  e.  V  ( 2nd `  ( 1st `  p ) )  =  x  /\  p  e.  ( ( V  X.  V )  X.  V
) )  /\  p  e.  ( V 2SPathOnOt  E )
) ) )
32 anass 649 . . . . . . . . 9  |-  ( ( ( p  e.  ( ( V  X.  V
)  X.  V )  /\  p  e.  ( V 2SPathOnOt  E ) )  /\  ( 2nd `  ( 1st `  p ) )  =  x )  <->  ( p  e.  ( ( V  X.  V )  X.  V
)  /\  ( p  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  p ) )  =  x ) ) )
3332bicomi 202 . . . . . . . 8  |-  ( ( p  e.  ( ( V  X.  V )  X.  V )  /\  ( p  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  p
) )  =  x ) )  <->  ( (
p  e.  ( ( V  X.  V )  X.  V )  /\  p  e.  ( V 2SPathOnOt  E ) )  /\  ( 2nd `  ( 1st `  p
) )  =  x ) )
3433rexbii 2965 . . . . . . 7  |-  ( E. x  e.  V  ( p  e.  ( ( V  X.  V )  X.  V )  /\  ( p  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  p
) )  =  x ) )  <->  E. x  e.  V  ( (
p  e.  ( ( V  X.  V )  X.  V )  /\  p  e.  ( V 2SPathOnOt  E ) )  /\  ( 2nd `  ( 1st `  p
) )  =  x ) )
35 ancom 450 . . . . . . . 8  |-  ( ( ( p  e.  ( ( V  X.  V
)  X.  V )  /\  p  e.  ( V 2SPathOnOt  E ) )  /\  E. x  e.  V  ( 2nd `  ( 1st `  p ) )  =  x )  <->  ( E. x  e.  V  ( 2nd `  ( 1st `  p
) )  =  x  /\  ( p  e.  ( ( V  X.  V )  X.  V
)  /\  p  e.  ( V 2SPathOnOt  E ) ) ) )
36 r19.42v 3016 . . . . . . . 8  |-  ( E. x  e.  V  ( ( p  e.  ( ( V  X.  V
)  X.  V )  /\  p  e.  ( V 2SPathOnOt  E ) )  /\  ( 2nd `  ( 1st `  p ) )  =  x )  <->  ( (
p  e.  ( ( V  X.  V )  X.  V )  /\  p  e.  ( V 2SPathOnOt  E ) )  /\  E. x  e.  V  ( 2nd `  ( 1st `  p
) )  =  x ) )
37 anass 649 . . . . . . . 8  |-  ( ( ( E. x  e.  V  ( 2nd `  ( 1st `  p ) )  =  x  /\  p  e.  ( ( V  X.  V )  X.  V
) )  /\  p  e.  ( V 2SPathOnOt  E )
)  <->  ( E. x  e.  V  ( 2nd `  ( 1st `  p
) )  =  x  /\  ( p  e.  ( ( V  X.  V )  X.  V
)  /\  p  e.  ( V 2SPathOnOt  E ) ) ) )
3835, 36, 373bitr4i 277 . . . . . . 7  |-  ( E. x  e.  V  ( ( p  e.  ( ( V  X.  V
)  X.  V )  /\  p  e.  ( V 2SPathOnOt  E ) )  /\  ( 2nd `  ( 1st `  p ) )  =  x )  <->  ( ( E. x  e.  V  ( 2nd `  ( 1st `  p ) )  =  x  /\  p  e.  ( ( V  X.  V )  X.  V
) )  /\  p  e.  ( V 2SPathOnOt  E )
) )
3934, 38bitri 249 . . . . . 6  |-  ( E. x  e.  V  ( p  e.  ( ( V  X.  V )  X.  V )  /\  ( p  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  p
) )  =  x ) )  <->  ( ( E. x  e.  V  ( 2nd `  ( 1st `  p ) )  =  x  /\  p  e.  ( ( V  X.  V )  X.  V
) )  /\  p  e.  ( V 2SPathOnOt  E )
) )
4031, 39syl6bbr 263 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  ->  (
p  e.  ( V 2SPathOnOt  E )  <->  E. x  e.  V  ( p  e.  ( ( V  X.  V )  X.  V
)  /\  ( p  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  p ) )  =  x ) ) ) )
41 simpr 461 . . . . . . . . 9  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  A. v  e.  V  ( ( V VDeg 
E ) `  v
)  =  K )  /\  x  e.  V
)  ->  x  e.  V )
42 3xpexg 6587 . . . . . . . . . . 11  |-  ( V  e.  Fin  ->  (
( V  X.  V
)  X.  V )  e.  _V )
43 rabexg 4597 . . . . . . . . . . 11  |-  ( ( ( V  X.  V
)  X.  V )  e.  _V  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) }  e.  _V )
4442, 43syl 16 . . . . . . . . . 10  |-  ( V  e.  Fin  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) }  e.  _V )
4544ad3antlr 730 . . . . . . . . 9  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  A. v  e.  V  ( ( V VDeg 
E ) `  v
)  =  K )  /\  x  e.  V
)  ->  { t  e.  ( ( V  X.  V )  X.  V
)  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  x ) }  e.  _V )
46 eqeq2 2482 . . . . . . . . . . . 12  |-  ( a  =  x  ->  (
( 2nd `  ( 1st `  t ) )  =  a  <->  ( 2nd `  ( 1st `  t
) )  =  x ) )
4746anbi2d 703 . . . . . . . . . . 11  |-  ( a  =  x  ->  (
( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  a )  <->  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  x ) ) )
4847rabbidv 3105 . . . . . . . . . 10  |-  ( a  =  x  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  a ) }  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) } )
49 usgreghash2spot.m . . . . . . . . . 10  |-  M  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) } )
5048, 49fvmptg 5948 . . . . . . . . 9  |-  ( ( x  e.  V  /\  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) }  e.  _V )  ->  ( M `  x )  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) } )
5141, 45, 50syl2anc 661 . . . . . . . 8  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  A. v  e.  V  ( ( V VDeg 
E ) `  v
)  =  K )  /\  x  e.  V
)  ->  ( M `  x )  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) } )
5251eleq2d 2537 . . . . . . 7  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  A. v  e.  V  ( ( V VDeg 
E ) `  v
)  =  K )  /\  x  e.  V
)  ->  ( p  e.  ( M `  x
)  <->  p  e.  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) } ) )
53 eleq1 2539 . . . . . . . . 9  |-  ( t  =  p  ->  (
t  e.  ( V 2SPathOnOt  E )  <->  p  e.  ( V 2SPathOnOt  E ) ) )
54 fveq2 5866 . . . . . . . . . . 11  |-  ( t  =  p  ->  ( 1st `  t )  =  ( 1st `  p
) )
5554fveq2d 5870 . . . . . . . . . 10  |-  ( t  =  p  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  p ) ) )
5655eqeq1d 2469 . . . . . . . . 9  |-  ( t  =  p  ->  (
( 2nd `  ( 1st `  t ) )  =  x  <->  ( 2nd `  ( 1st `  p
) )  =  x ) )
5753, 56anbi12d 710 . . . . . . . 8  |-  ( t  =  p  ->  (
( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x )  <->  ( p  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  p ) )  =  x ) ) )
5857elrab 3261 . . . . . . 7  |-  ( p  e.  { t  e.  ( ( V  X.  V )  X.  V
)  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  x ) }  <-> 
( p  e.  ( ( V  X.  V
)  X.  V )  /\  ( p  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  p ) )  =  x ) ) )
5952, 58syl6rbb 262 . . . . . 6  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  A. v  e.  V  ( ( V VDeg 
E ) `  v
)  =  K )  /\  x  e.  V
)  ->  ( (
p  e.  ( ( V  X.  V )  X.  V )  /\  ( p  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  p
) )  =  x ) )  <->  p  e.  ( M `  x ) ) )
6059rexbidva 2970 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  ->  ( E. x  e.  V  ( p  e.  (
( V  X.  V
)  X.  V )  /\  ( p  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  p ) )  =  x ) )  <->  E. x  e.  V  p  e.  ( M `  x ) ) )
6140, 60bitrd 253 . . . 4  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  ->  (
p  e.  ( V 2SPathOnOt  E )  <->  E. x  e.  V  p  e.  ( M `  x ) ) )
62 eliun 4330 . . . 4  |-  ( p  e.  U_ x  e.  V  ( M `  x )  <->  E. x  e.  V  p  e.  ( M `  x ) )
6361, 62syl6bbr 263 . . 3  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  ->  (
p  e.  ( V 2SPathOnOt  E )  <->  p  e.  U_ x  e.  V  ( M `  x ) ) )
6463eqrdv 2464 . 2  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  ->  ( V 2SPathOnOt  E )  =  U_ x  e.  V  ( M `  x )
)
6564ex 434 1  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  ( A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K  ->  ( V 2SPathOnOt  E )  =  U_ x  e.  V  ( M `  x )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113   <.cotp 4035   U_ciun 4325   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   ` cfv 5588  (class class class)co 6284   1stc1st 6782   2ndc2nd 6783   Fincfn 7516   0cc0 9492   1c1 9493   2c2 10585   #chash 12373   USGrph cusg 24034   SPaths cspath 24205   2SPathOnOt c2spthot 24560   VDeg cvdg 24597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-usgra 24037  df-wlk 24212  df-trail 24213  df-pth 24214  df-spth 24215  df-wlkon 24218  df-spthon 24221  df-2spthonot 24564  df-2spthsot 24565
This theorem is referenced by:  usgreghash2spot  24774
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