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Theorem usgreg2spot 25793
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 25063 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 el2pthsot 25607 . . . . . . . . . 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 17 . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  y  e.  V )  ->  y  e.  V )
54ad2antrr 730 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  y  e.  V )
6 simpr 462 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( V USGrph  E  /\  y  e.  V )  /\  x  e.  V
)  ->  x  e.  V )
76adantr 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  x  e.  V )
8 simpr 462 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  z  e.  V )
95, 7, 83jca 1185 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  (
y  e.  V  /\  x  e.  V  /\  z  e.  V )
)
10 ot2ndg 6822 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  V  /\  x  e.  V  /\  z  e.  V )  ->  ( 2nd `  ( 1st `  <. y ,  x ,  z >. )
)  =  x )
119, 10syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  ( 2nd `  ( 1st `  <. y ,  x ,  z
>. ) )  =  x )
12 eqidd 2423 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  <. y ,  x ,  z >.  =  <. y ,  x ,  z >. )
13 otel3xp 4889 . . . . . . . . . . . . . . . . . 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 665 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  <. y ,  x ,  z >.  e.  ( ( V  X.  V )  X.  V
) )
1511, 14jca 534 . . . . . . . . . . . . . . . 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 5881 . . . . . . . . . . . . . . . . . . 19  |-  ( p  =  <. y ,  x ,  z >.  ->  ( 1st `  p )  =  ( 1st `  <. y ,  x ,  z
>. ) )
1716fveq2d 5885 . . . . . . . . . . . . . . . . . 18  |-  ( p  =  <. y ,  x ,  z >.  ->  ( 2nd `  ( 1st `  p
) )  =  ( 2nd `  ( 1st `  <. y ,  x ,  z >. )
) )
1817eqeq1d 2424 . . . . . . . . . . . . . . . . 17  |-  ( p  =  <. y ,  x ,  z >.  ->  (
( 2nd `  ( 1st `  p ) )  =  x  <->  ( 2nd `  ( 1st `  <. y ,  x ,  z
>. ) )  =  x ) )
19 eleq1 2495 . . . . . . . . . . . . . . . . 17  |-  ( p  =  <. y ,  x ,  z >.  ->  (
p  e.  ( ( V  X.  V )  X.  V )  <->  <. y ,  x ,  z >.  e.  ( ( V  X.  V )  X.  V
) ) )
2018, 19anbi12d 715 . . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . 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 32 . . . . . . . . . . . . 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 2914 . . . . . . . . . . . 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 2897 . . . . . . . . . . 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 2977 . . . . . . . . . . 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 229 . . . . . . . . . 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 2914 . . . . . . . . 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 218 . . . . . . . 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 730 . . . . . . 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 639 . . . . . 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 653 . . . . . . . . 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 205 . . . . . . . 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 2924 . . . . . . 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 451 . . . . . . . 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 2980 . . . . . . . 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 653 . . . . . . . 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 280 . . . . . . 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 252 . . . . . 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 266 . . . . 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 462 . . . . . . . . 9  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  A. v  e.  V  ( ( V VDeg 
E ) `  v
)  =  K )  /\  x  e.  V
)  ->  x  e.  V )
42 3xpexg 6608 . . . . . . . . . . 11  |-  ( V  e.  Fin  ->  (
( V  X.  V
)  X.  V )  e.  _V )
43 rabexg 4574 . . . . . . . . . . 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 17 . . . . . . . . . 10  |-  ( V  e.  Fin  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x ) }  e.  _V )
4544ad3antlr 735 . . . . . . . . 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 2437 . . . . . . . . . . . 12  |-  ( a  =  x  ->  (
( 2nd `  ( 1st `  t ) )  =  a  <->  ( 2nd `  ( 1st `  t
) )  =  x ) )
4746anbi2d 708 . . . . . . . . . . 11  |-  ( a  =  x  ->  (
( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  a )  <->  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  x ) ) )
4847rabbidv 3071 . . . . . . . . . 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 5962 . . . . . . . . 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 665 . . . . . . . 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 2492 . . . . . . 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 2495 . . . . . . . . 9  |-  ( t  =  p  ->  (
t  e.  ( V 2SPathOnOt  E )  <->  p  e.  ( V 2SPathOnOt  E ) ) )
54 fveq2 5881 . . . . . . . . . . 11  |-  ( t  =  p  ->  ( 1st `  t )  =  ( 1st `  p
) )
5554fveq2d 5885 . . . . . . . . . 10  |-  ( t  =  p  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  p ) ) )
5655eqeq1d 2424 . . . . . . . . 9  |-  ( t  =  p  ->  (
( 2nd `  ( 1st `  t ) )  =  x  <->  ( 2nd `  ( 1st `  p
) )  =  x ) )
5753, 56anbi12d 715 . . . . . . . 8  |-  ( t  =  p  ->  (
( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x )  <->  ( p  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  p ) )  =  x ) ) )
5857elrab 3228 . . . . . . 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 265 . . . . . 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 2933 . . . . 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 256 . . . 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 4304 . . . 4  |-  ( p  e.  U_ x  e.  V  ( M `  x )  <->  E. x  e.  V  p  e.  ( M `  x ) )
6361, 62syl6bbr 266 . . 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 2419 . 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 435 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 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872   A.wral 2771   E.wrex 2772   {crab 2775   _Vcvv 3080   <.cotp 4006   U_ciun 4299   class class class wbr 4423    |-> cmpt 4482    X. cxp 4851   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   Fincfn 7580   0cc0 9546   1c1 9547   2c2 10666   #chash 12521   USGrph cusg 25055   SPaths cspath 25227   2SPathOnOt c2spthot 25582   VDeg cvdg 25619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-ot 4007  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12522  df-word 12668  df-usgra 25058  df-wlk 25234  df-trail 25235  df-pth 25236  df-spth 25237  df-wlkon 25240  df-spthon 25243  df-2spthonot 25586  df-2spthsot 25587
This theorem is referenced by:  usgreghash2spot  25795
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