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Theorem usgreg2spot 25188
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 24459 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 el2pthsot 25002 . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  y  e.  V )  ->  y  e.  V )
54ad2antrr 723 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  y  e.  V )
6 simpr 459 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( V USGrph  E  /\  y  e.  V )  /\  x  e.  V
)  ->  x  e.  V )
76adantr 463 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  x  e.  V )
8 simpr 459 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  z  e.  V )
95, 7, 83jca 1174 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  (
y  e.  V  /\  x  e.  V  /\  z  e.  V )
)
10 ot2ndg 6714 . . . . . . . . . . . . . . . . . 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 2383 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  <. y ,  x ,  z >.  =  <. y ,  x ,  z >. )
13 otel3xp 4949 . . . . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( V USGrph  E  /\  y  e.  V
)  /\  x  e.  V )  /\  z  e.  V )  ->  <. y ,  x ,  z >.  e.  ( ( V  X.  V )  X.  V
) )
1511, 14jca 530 . . . . . . . . . . . . . . . 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 5774 . . . . . . . . . . . . . . . . . . 19  |-  ( p  =  <. y ,  x ,  z >.  ->  ( 1st `  p )  =  ( 1st `  <. y ,  x ,  z
>. ) )
1716fveq2d 5778 . . . . . . . . . . . . . . . . . 18  |-  ( p  =  <. y ,  x ,  z >.  ->  ( 2nd `  ( 1st `  p
) )  =  ( 2nd `  ( 1st `  <. y ,  x ,  z >. )
) )
1817eqeq1d 2384 . . . . . . . . . . . . . . . . 17  |-  ( p  =  <. y ,  x ,  z >.  ->  (
( 2nd `  ( 1st `  p ) )  =  x  <->  ( 2nd `  ( 1st `  <. y ,  x ,  z
>. ) )  =  x ) )
19 eleq1 2454 . . . . . . . . . . . . . . . . 17  |-  ( p  =  <. y ,  x ,  z >.  ->  (
p  e.  ( ( V  X.  V )  X.  V )  <->  <. y ,  x ,  z >.  e.  ( ( V  X.  V )  X.  V
) ) )
2018, 19anbi12d 708 . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . 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 2874 . . . . . . . . . . . 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 2857 . . . . . . . . . . 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 2934 . . . . . . . . . . 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 2874 . . . . . . . . 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 723 . . . . . . 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 633 . . . . . 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 647 . . . . . . . . 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 2884 . . . . . . 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 448 . . . . . . . 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 2937 . . . . . . . 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 647 . . . . . . . 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 459 . . . . . . . . 9  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  A. v  e.  V  ( ( V VDeg 
E ) `  v
)  =  K )  /\  x  e.  V
)  ->  x  e.  V )
42 3xpexg 6502 . . . . . . . . . . 11  |-  ( V  e.  Fin  ->  (
( V  X.  V
)  X.  V )  e.  _V )
43 rabexg 4515 . . . . . . . . . . 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 728 . . . . . . . . 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 2397 . . . . . . . . . . . 12  |-  ( a  =  x  ->  (
( 2nd `  ( 1st `  t ) )  =  a  <->  ( 2nd `  ( 1st `  t
) )  =  x ) )
4746anbi2d 701 . . . . . . . . . . 11  |-  ( a  =  x  ->  (
( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  a )  <->  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  x ) ) )
4847rabbidv 3026 . . . . . . . . . 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 5855 . . . . . . . . 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 659 . . . . . . . 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 2452 . . . . . . 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 2454 . . . . . . . . 9  |-  ( t  =  p  ->  (
t  e.  ( V 2SPathOnOt  E )  <->  p  e.  ( V 2SPathOnOt  E ) ) )
54 fveq2 5774 . . . . . . . . . . 11  |-  ( t  =  p  ->  ( 1st `  t )  =  ( 1st `  p
) )
5554fveq2d 5778 . . . . . . . . . 10  |-  ( t  =  p  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  p ) ) )
5655eqeq1d 2384 . . . . . . . . 9  |-  ( t  =  p  ->  (
( 2nd `  ( 1st `  t ) )  =  x  <->  ( 2nd `  ( 1st `  p
) )  =  x ) )
5753, 56anbi12d 708 . . . . . . . 8  |-  ( t  =  p  ->  (
( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  x )  <->  ( p  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  p ) )  =  x ) ) )
5857elrab 3182 . . . . . . 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 2890 . . . . 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 4248 . . . 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 2379 . 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 432 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 367    /\ w3a 971    = wceq 1399   E.wex 1620    e. wcel 1826   A.wral 2732   E.wrex 2733   {crab 2736   _Vcvv 3034   <.cotp 3952   U_ciun 4243   class class class wbr 4367    |-> cmpt 4425    X. cxp 4911   ` cfv 5496  (class class class)co 6196   1stc1st 6697   2ndc2nd 6698   Fincfn 7435   0cc0 9403   1c1 9404   2c2 10502   #chash 12307   USGrph cusg 24451   SPaths cspath 24622   2SPathOnOt c2spthot 24977   VDeg cvdg 25014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
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 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-ot 3953  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-hash 12308  df-word 12446  df-usgra 24454  df-wlk 24629  df-trail 24630  df-pth 24631  df-spth 24632  df-wlkon 24635  df-spthon 24638  df-2spthonot 24981  df-2spthsot 24982
This theorem is referenced by:  usgreghash2spot  25190
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