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Theorem 4cycl4dv4e 28414
Description: If there is a cycle of length 4 in a graph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
Assertion
Ref Expression
4cycl4dv4e  |-  ( ( V USGrph  E  /\  F ( V Cycles  E ) P  /\  ( # `  F
)  =  4 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )
Distinct variable groups:    E, a,
b, c, d    P, a, b, c, d    V, a, b, c, d
Allowed substitution hints:    F( a, b, c, d)

Proof of Theorem 4cycl4dv4e
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 cycliswlk 28377 . . . . 5  |-  ( F ( V Cycles  E ) P  ->  F ( V Walks  E ) P )
2 wlkbprop 28333 . . . . 5  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
31, 2syl 15 . . . 4  |-  ( F ( V Cycles  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
4 iscycl 28370 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Cycles  E ) P 
<->  ( F ( V Paths 
E ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
5 ispth 28354 . . . . . . . 8  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Paths  E ) P 
<->  ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) ) ) )
6 istrl 28336 . . . . . . . . . 10  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Trails  E ) P 
<->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
7 fzo0to42pr 28211 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0..^ 4 )  =  ( { 0 ,  1 }  u.  { 2 ,  3 } )
87raleqi 2753 . . . . . . . . . . . . . . . . . . 19  |-  ( A. k  e.  ( 0..^ 4 ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  A. k  e.  ( { 0 ,  1 }  u.  { 2 ,  3 } ) ( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
9 ralunb 3369 . . . . . . . . . . . . . . . . . . 19  |-  ( A. k  e.  ( {
0 ,  1 }  u.  { 2 ,  3 } ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  <->  ( A. k  e.  { 0 ,  1 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  /\  A. k  e.  { 2 ,  3 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
10 0z 10051 . . . . . . . . . . . . . . . . . . . . 21  |-  0  e.  ZZ
11 1z 10069 . . . . . . . . . . . . . . . . . . . . 21  |-  1  e.  ZZ
12 fveq2 5541 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
1312fveq2d 5545 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  0  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  0 )
) )
14 fveq2 5541 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  0  ->  ( P `  k )  =  ( P ` 
0 ) )
15 oveq1 5881 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( k  =  0  ->  (
k  +  1 )  =  ( 0  +  1 ) )
16 0p1e1 9855 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 0  +  1 )  =  1
1715, 16syl6eq 2344 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( k  =  0  ->  (
k  +  1 )  =  1 )
1817fveq2d 5545 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  0  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
1 ) )
1914, 18preq12d 3727 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  0  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
0 ) ,  ( P `  1 ) } )
2013, 19eqeq12d 2310 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  0  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } ) )
21 fveq2 5541 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  1  ->  ( F `  k )  =  ( F ` 
1 ) )
2221fveq2d 5545 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  1  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  1 )
) )
23 fveq2 5541 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  1  ->  ( P `  k )  =  ( P ` 
1 ) )
24 oveq1 5881 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( k  =  1  ->  (
k  +  1 )  =  ( 1  +  1 ) )
25 1p1e2 9856 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 1  +  1 )  =  2
2624, 25syl6eq 2344 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( k  =  1  ->  (
k  +  1 )  =  2 )
2726fveq2d 5545 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  1  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
2 ) )
2823, 27preq12d 3727 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  1  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
1 ) ,  ( P `  2 ) } )
2922, 28eqeq12d 2310 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  1  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } ) )
3020, 29ralprg 3695 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 0  e.  ZZ  /\  1  e.  ZZ )  ->  ( A. k  e. 
{ 0 ,  1 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( ( E `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) )
3110, 11, 30mp2an 653 . . . . . . . . . . . . . . . . . . . 20  |-  ( A. k  e.  { 0 ,  1 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) )
32 2z 10070 . . . . . . . . . . . . . . . . . . . . 21  |-  2  e.  ZZ
33 3nn0 9999 . . . . . . . . . . . . . . . . . . . . . 22  |-  3  e.  NN0
3433nn0zi 10064 . . . . . . . . . . . . . . . . . . . . 21  |-  3  e.  ZZ
35 fveq2 5541 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  2  ->  ( F `  k )  =  ( F ` 
2 ) )
3635fveq2d 5545 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  2  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  2 )
) )
37 fveq2 5541 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  2  ->  ( P `  k )  =  ( P ` 
2 ) )
38 oveq1 5881 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( k  =  2  ->  (
k  +  1 )  =  ( 2  +  1 ) )
39 2p1e3 9863 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 2  +  1 )  =  3
4038, 39syl6eq 2344 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( k  =  2  ->  (
k  +  1 )  =  3 )
4140fveq2d 5545 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  2  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
3 ) )
4237, 41preq12d 3727 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  2  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
2 ) ,  ( P `  3 ) } )
4336, 42eqeq12d 2310 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  2  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  2
) )  =  {
( P `  2
) ,  ( P `
 3 ) } ) )
44 fveq2 5541 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  3  ->  ( F `  k )  =  ( F ` 
3 ) )
4544fveq2d 5545 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  3  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  3 )
) )
46 fveq2 5541 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  3  ->  ( P `  k )  =  ( P ` 
3 ) )
47 oveq1 5881 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( k  =  3  ->  (
k  +  1 )  =  ( 3  +  1 ) )
48 3p1e4 9864 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 3  +  1 )  =  4
4947, 48syl6eq 2344 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( k  =  3  ->  (
k  +  1 )  =  4 )
5049fveq2d 5545 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  3  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
4 ) )
5146, 50preq12d 3727 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  3  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
3 ) ,  ( P `  4 ) } )
5245, 51eqeq12d 2310 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  3  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  3
) )  =  {
( P `  3
) ,  ( P `
 4 ) } ) )
5343, 52ralprg 3695 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 2  e.  ZZ  /\  3  e.  ZZ )  ->  ( A. k  e. 
{ 2 ,  3 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( ( E `  ( F `  2 ) )  =  { ( P `
 2 ) ,  ( P `  3
) }  /\  ( E `  ( F `  3 ) )  =  { ( P `
 3 ) ,  ( P `  4
) } ) ) )
5432, 34, 53mp2an 653 . . . . . . . . . . . . . . . . . . . 20  |-  ( A. k  e.  { 2 ,  3 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( ( E `
 ( F ` 
2 ) )  =  { ( P ` 
2 ) ,  ( P `  3 ) }  /\  ( E `
 ( F ` 
3 ) )  =  { ( P ` 
3 ) ,  ( P `  4 ) } ) )
5531, 54anbi12i 678 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A. k  e.  {
0 ,  1 }  ( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  /\  A. k  e.  { 2 ,  3 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  <->  ( (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )  /\  ( ( E `  ( F `  2 ) )  =  { ( P `  2 ) ,  ( P ` 
3 ) }  /\  ( E `  ( F `
 3 ) )  =  { ( P `
 3 ) ,  ( P `  4
) } ) ) )
568, 9, 553bitri 262 . . . . . . . . . . . . . . . . . 18  |-  ( A. k  e.  ( 0..^ 4 ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( ( ( E `  ( F `
 0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )  /\  ( ( E `  ( F `  2 ) )  =  { ( P `  2 ) ,  ( P ` 
3 ) }  /\  ( E `  ( F `
 3 ) )  =  { ( P `
 3 ) ,  ( P `  4
) } ) ) )
57 preq2 3720 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( P `  4 )  =  ( P ` 
0 )  ->  { ( P `  3 ) ,  ( P ` 
4 ) }  =  { ( P ` 
3 ) ,  ( P `  0 ) } )
5857eqcoms 2299 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( P `  0 )  =  ( P ` 
4 )  ->  { ( P `  3 ) ,  ( P ` 
4 ) }  =  { ( P ` 
3 ) ,  ( P `  0 ) } )
5958eqeq2d 2307 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( P `  0 )  =  ( P ` 
4 )  ->  (
( E `  ( F `  3 )
)  =  { ( P `  3 ) ,  ( P ` 
4 ) }  <->  ( E `  ( F `  3
) )  =  {
( P `  3
) ,  ( P `
 0 ) } ) )
6059anbi2d 684 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( P `  0 )  =  ( P ` 
4 )  ->  (
( ( E `  ( F `  2 ) )  =  { ( P `  2 ) ,  ( P ` 
3 ) }  /\  ( E `  ( F `
 3 ) )  =  { ( P `
 3 ) ,  ( P `  4
) } )  <->  ( ( E `  ( F `  2 ) )  =  { ( P `
 2 ) ,  ( P `  3
) }  /\  ( E `  ( F `  3 ) )  =  { ( P `
 3 ) ,  ( P `  0
) } ) ) )
6160anbi2d 684 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( P `  0 )  =  ( P ` 
4 )  ->  (
( ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } )  /\  (
( E `  ( F `  2 )
)  =  { ( P `  2 ) ,  ( P ` 
3 ) }  /\  ( E `  ( F `
 3 ) )  =  { ( P `
 3 ) ,  ( P `  4
) } ) )  <-> 
( ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } )  /\  (
( E `  ( F `  2 )
)  =  { ( P `  2 ) ,  ( P ` 
3 ) }  /\  ( E `  ( F `
 3 ) )  =  { ( P `
 3 ) ,  ( P `  0
) } ) ) ) )
62 simpll 730 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F ) )  /\  ( # `  F )  =  4 )  ->  V USGrph  E )
63 simplrl 736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F ) )  /\  ( # `  F )  =  4 )  ->  F  e. Word  dom 
E )
64 simplrr 737 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F ) )  /\  ( # `  F )  =  4 )  ->  Fun  `' F
)
65 simpr 447 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F ) )  /\  ( # `  F )  =  4 )  ->  ( # `  F
)  =  4 )
66 4cycl4dv 28413 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `
 F )  =  4 ) )  -> 
( ( ( ( E `  ( F `
 0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )  /\  ( ( E `  ( F `  2 ) )  =  { ( P `  2 ) ,  ( P ` 
3 ) }  /\  ( E `  ( F `
 3 ) )  =  { ( P `
 3 ) ,  ( P `  0
) } ) )  ->  ( ( ( { ( P ` 
0 ) ,  ( P `  1 ) }  e.  ran  E  /\  { ( P ` 
1 ) ,  ( P `  2 ) }  e.  ran  E
)  /\  ( {
( P `  2
) ,  ( P `
 3 ) }  e.  ran  E  /\  { ( P `  3
) ,  ( P `
 0 ) }  e.  ran  E ) )  /\  ( ( ( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  0 )  =/=  ( P `  3
) )  /\  (
( P `  1
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
3 )  /\  ( P `  2 )  =/=  ( P `  3
) ) ) ) ) )
6762, 63, 64, 65, 66syl13anc 1184 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F ) )  /\  ( # `  F )  =  4 )  ->  ( (
( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )  /\  ( ( E `  ( F `  2 ) )  =  { ( P `  2 ) ,  ( P ` 
3 ) }  /\  ( E `  ( F `
 3 ) )  =  { ( P `
 3 ) ,  ( P `  0
) } ) )  ->  ( ( ( { ( P ` 
0 ) ,  ( P `  1 ) }  e.  ran  E  /\  { ( P ` 
1 ) ,  ( P `  2 ) }  e.  ran  E
)  /\  ( {
( P `  2
) ,  ( P `
 3 ) }  e.  ran  E  /\  { ( P `  3
) ,  ( P `
 0 ) }  e.  ran  E ) )  /\  ( ( ( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  0 )  =/=  ( P `  3
) )  /\  (
( P `  1
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
3 )  /\  ( P `  2 )  =/=  ( P `  3
) ) ) ) ) )
6867imp 418 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F
) )  /\  ( # `
 F )  =  4 )  /\  (
( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )  /\  ( ( E `  ( F `  2 ) )  =  { ( P `  2 ) ,  ( P ` 
3 ) }  /\  ( E `  ( F `
 3 ) )  =  { ( P `
 3 ) ,  ( P `  0
) } ) ) )  ->  ( (
( { ( P `
 0 ) ,  ( P `  1
) }  e.  ran  E  /\  { ( P `
 1 ) ,  ( P `  2
) }  e.  ran  E )  /\  ( { ( P `  2
) ,  ( P `
 3 ) }  e.  ran  E  /\  { ( P `  3
) ,  ( P `
 0 ) }  e.  ran  E ) )  /\  ( ( ( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  0 )  =/=  ( P `  3
) )  /\  (
( P `  1
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
3 )  /\  ( P `  2 )  =/=  ( P `  3
) ) ) ) )
69 4nn0 10000 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  4  e.  NN0
7069nn0zi 10064 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  4  e.  ZZ
71 3re 9833 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  3  e.  RR
72 4re 9835 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  4  e.  RR
73 3lt4 9905 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  3  <  4
7471, 72, 73ltleii 8957 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  3  <_  4
75 eluz2 10252 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( 4  e.  ( ZZ>= `  3
)  <->  ( 3  e.  ZZ  /\  4  e.  ZZ  /\  3  <_ 
4 ) )
7634, 70, 74, 75mpbir3an 1134 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  4  e.  ( ZZ>= `  3 )
77 4fvwrd4 28220 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( 4  e.  ( ZZ>= ` 
3 )  /\  P : ( 0 ... 4 ) --> V )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( P `
 0 )  =  a  /\  ( P `
 1 )  =  b )  /\  (
( P `  2
)  =  c  /\  ( P `  3 )  =  d ) ) )
7876, 77mpan 651 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( P : ( 0 ... 4 ) --> V  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( P `
 0 )  =  a  /\  ( P `
 1 )  =  b )  /\  (
( P `  2
)  =  c  /\  ( P `  3 )  =  d ) ) )
79 preq12 3721 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( P `  0
)  =  a  /\  ( P `  1 )  =  b )  ->  { ( P ` 
0 ) ,  ( P `  1 ) }  =  { a ,  b } )
8079adantr 451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  { ( P `
 0 ) ,  ( P `  1
) }  =  {
a ,  b } )
8180eleq1d 2362 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( { ( P `  0 ) ,  ( P ` 
1 ) }  e.  ran  E  <->  { a ,  b }  e.  ran  E
) )
82 simplr 731 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( P ` 
1 )  =  b )
83 simprl 732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( P ` 
2 )  =  c )
8482, 83preq12d 3727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  { ( P `
 1 ) ,  ( P `  2
) }  =  {
b ,  c } )
8584eleq1d 2362 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( { ( P `  1 ) ,  ( P ` 
2 ) }  e.  ran  E  <->  { b ,  c }  e.  ran  E
) )
8681, 85anbi12d 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( { ( P `  0
) ,  ( P `
 1 ) }  e.  ran  E  /\  { ( P `  1
) ,  ( P `
 2 ) }  e.  ran  E )  <-> 
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) ) )
87 preq12 3721 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( P `  2
)  =  c  /\  ( P `  3 )  =  d )  ->  { ( P ` 
2 ) ,  ( P `  3 ) }  =  { c ,  d } )
8887adantl 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  { ( P `
 2 ) ,  ( P `  3
) }  =  {
c ,  d } )
8988eleq1d 2362 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( { ( P `  2 ) ,  ( P ` 
3 ) }  e.  ran  E  <->  { c ,  d }  e.  ran  E
) )
90 simprr 733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( P ` 
3 )  =  d )
91 simpll 730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( P ` 
0 )  =  a )
9290, 91preq12d 3727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  { ( P `
 3 ) ,  ( P `  0
) }  =  {
d ,  a } )
9392eleq1d 2362 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( { ( P `  3 ) ,  ( P ` 
0 ) }  e.  ran  E  <->  { d ,  a }  e.  ran  E
) )
9489, 93anbi12d 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( { ( P `  2
) ,  ( P `
 3 ) }  e.  ran  E  /\  { ( P `  3
) ,  ( P `
 0 ) }  e.  ran  E )  <-> 
( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) ) )
9586, 94anbi12d 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( ( { ( P ` 
0 ) ,  ( P `  1 ) }  e.  ran  E  /\  { ( P ` 
1 ) ,  ( P `  2 ) }  e.  ran  E
)  /\  ( {
( P `  2
) ,  ( P `
 3 ) }  e.  ran  E  /\  { ( P `  3
) ,  ( P `
 0 ) }  e.  ran  E ) )  <->  ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) ) ) )
9691, 82neeq12d 2474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( P `
 0 )  =/=  ( P `  1
)  <->  a  =/=  b
) )
9791, 83neeq12d 2474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( P `
 0 )  =/=  ( P `  2
)  <->  a  =/=  c
) )
9891, 90neeq12d 2474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( P `
 0 )  =/=  ( P `  3
)  <->  a  =/=  d
) )
9996, 97, 983anbi123d 1252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( ( P `  0 )  =/=  ( P ` 
1 )  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  0 )  =/=  ( P `  3
) )  <->  ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d ) ) )
10082, 83neeq12d 2474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( P `
 1 )  =/=  ( P `  2
)  <->  b  =/=  c
) )
10182, 90neeq12d 2474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( P `
 1 )  =/=  ( P `  3
)  <->  b  =/=  d
) )
102 simpl 443 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( P `  2
)  =  c  /\  ( P `  3 )  =  d )  -> 
( P `  2
)  =  c )
103 simpr 447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( P `  2
)  =  c  /\  ( P `  3 )  =  d )  -> 
( P `  3
)  =  d )
104102, 103neeq12d 2474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( P `  2
)  =  c  /\  ( P `  3 )  =  d )  -> 
( ( P ` 
2 )  =/=  ( P `  3 )  <->  c  =/=  d ) )
105104adantl 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( P `
 2 )  =/=  ( P `  3
)  <->  c  =/=  d
) )
106100, 101, 1053anbi123d 1252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( ( P `  1 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  3
)  /\  ( P `  2 )  =/=  ( P `  3
) )  <->  ( b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) )
10799, 106anbi12d 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( ( ( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  0 )  =/=  ( P `  3
) )  /\  (
( P `  1
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
3 )  /\  ( P `  2 )  =/=  ( P `  3
) ) )  <->  ( (
a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )
10895, 107anbi12d 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( ( ( { ( P `
 0 ) ,  ( P `  1
) }  e.  ran  E  /\  { ( P `
 1 ) ,  ( P `  2
) }  e.  ran  E )  /\  ( { ( P `  2
) ,  ( P `
 3 ) }  e.  ran  E  /\  { ( P `  3
) ,  ( P `
 0 ) }  e.  ran  E ) )  /\  ( ( ( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  0 )  =/=  ( P `  3
) )  /\  (
( P `  1
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
3 )  /\  ( P `  2 )  =/=  ( P `  3
) ) ) )  <-> 
( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) )
109108biimpcd 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( ( { ( P `  0 ) ,  ( P ` 
1 ) }  e.  ran  E  /\  { ( P `  1 ) ,  ( P ` 
2 ) }  e.  ran  E )  /\  ( { ( P ` 
2 ) ,  ( P `  3 ) }  e.  ran  E  /\  { ( P ` 
3 ) ,  ( P `  0 ) }  e.  ran  E
) )  /\  (
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  0 )  =/=  ( P ` 
3 ) )  /\  ( ( P ` 
1 )  =/=  ( P `  2 )  /\  ( P `  1
)  =/=  ( P `
 3 )  /\  ( P `  2 )  =/=  ( P ` 
3 ) ) ) )  ->  ( (
( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  /\  ( {
c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) )
110109reximdv 2667 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( { ( P `  0 ) ,  ( P ` 
1 ) }  e.  ran  E  /\  { ( P `  1 ) ,  ( P ` 
2 ) }  e.  ran  E )  /\  ( { ( P ` 
2 ) ,  ( P `  3 ) }  e.  ran  E  /\  { ( P ` 
3 ) ,  ( P `  0 ) }  e.  ran  E
) )  /\  (
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  0 )  =/=  ( P ` 
3 ) )  /\  ( ( P ` 
1 )  =/=  ( P `  2 )  /\  ( P `  1
)  =/=  ( P `
 3 )  /\  ( P `  2 )  =/=  ( P ` 
3 ) ) ) )  ->  ( E. d  e.  V  (
( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) )
111110reximdv 2667 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( ( { ( P `  0 ) ,  ( P ` 
1 ) }  e.  ran  E  /\  { ( P `  1 ) ,  ( P ` 
2 ) }  e.  ran  E )  /\  ( { ( P ` 
2 ) ,  ( P `  3 ) }  e.  ran  E  /\  { ( P ` 
3 ) ,  ( P `  0 ) }  e.  ran  E
) )  /\  (
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  0 )  =/=  ( P ` 
3 ) )  /\  ( ( P ` 
1 )  =/=  ( P `  2 )  /\  ( P `  1
)  =/=  ( P `
 3 )  /\  ( P `  2 )  =/=  ( P ` 
3 ) ) ) )  ->  ( E. c  e.  V  E. d  e.  V  (
( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) )
112111reximdv 2667 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( { ( P `  0 ) ,  ( P ` 
1 ) }  e.  ran  E  /\  { ( P `  1 ) ,  ( P ` 
2 ) }  e.  ran  E )  /\  ( { ( P ` 
2 ) ,  ( P `  3 ) }  e.  ran  E  /\  { ( P ` 
3 ) ,  ( P `  0 ) }  e.  ran  E
) )  /\  (
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  0 )  =/=  ( P ` 
3 ) )  /\  ( ( P ` 
1 )  =/=  ( P `  2 )  /\  ( P `  1
)  =/=  ( P `
 3 )  /\  ( P `  2 )  =/=  ( P ` 
3 ) ) ) )  ->  ( E. b  e.  V  E. c  e.  V  E. d  e.  V  (
( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) )
113112reximdv 2667 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( { ( P `  0 ) ,  ( P ` 
1 ) }  e.  ran  E  /\  { ( P `  1 ) ,  ( P ` 
2 ) }  e.  ran  E )  /\  ( { ( P ` 
2 ) ,  ( P `  3 ) }  e.  ran  E  /\  { ( P ` 
3 ) ,  ( P `  0 ) }  e.  ran  E
) )  /\  (
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  0 )  =/=  ( P ` 
3 ) )  /\  ( ( P ` 
1 )  =/=  ( P `  2 )  /\  ( P `  1
)  =/=  ( P `
 3 )  /\  ( P `  2 )  =/=  ( P ` 
3 ) ) ) )  ->  ( E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  (
( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) )
11468, 78, 113syl2im 34 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F
) )  /\  ( # `
 F )  =  4 )  /\  (
( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )  /\  ( ( E `  ( F `  2 ) )  =  { ( P `  2 ) ,  ( P ` 
3 ) }  /\  ( E `  ( F `
 3 ) )  =  { ( P `
 3 ) ,  ( P `  0
) } ) ) )  ->  ( P : ( 0 ... 4 ) --> V  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) )
115114exp41 593 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( V USGrph  E  ->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  ->  (
( # `  F )  =  4  ->  (
( ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } )  /\  (
( E `  ( F `  2 )
)  =  { ( P `  2 ) ,  ( P ` 
3 ) }  /\  ( E `  ( F `
 3 ) )  =  { ( P `
 3 ) ,  ( P `  0
) } ) )  ->  ( P :
( 0 ... 4
) --> V  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) ) )
116115com14 82 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )  /\  ( ( E `  ( F `  2 ) )  =  { ( P `  2 ) ,  ( P ` 
3 ) }  /\  ( E `  ( F `
 3 ) )  =  { ( P `
 3 ) ,  ( P `  0
) } ) )  ->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  ->  (
( # `  F )  =  4  ->  ( V USGrph  E  ->  ( P : ( 0 ... 4 ) --> V  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) ) )
117116com35 84 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )  /\  ( ( E `  ( F `  2 ) )  =  { ( P `  2 ) ,  ( P ` 
3 ) }  /\  ( E `  ( F `
 3 ) )  =  { ( P `
 3 ) ,  ( P `  0
) } ) )  ->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  ->  ( P : ( 0 ... 4 ) --> V  -> 
( V USGrph  E  ->  ( ( # `  F
)  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) ) )
11861, 117syl6bi 219 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( P `  0 )  =  ( P ` 
4 )  ->  (
( ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } )  /\  (
( E `  ( F `  2 )
)  =  { ( P `  2 ) ,  ( P ` 
3 ) }  /\  ( E `  ( F `
 3 ) )  =  { ( P `
 3 ) ,  ( P `  4
) } ) )  ->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  ->  ( P : ( 0 ... 4 ) --> V  -> 
( V USGrph  E  ->  ( ( # `  F
)  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) ) ) )
119118com12 27 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )  /\  ( ( E `  ( F `  2 ) )  =  { ( P `  2 ) ,  ( P ` 
3 ) }  /\  ( E `  ( F `
 3 ) )  =  { ( P `
 3 ) ,  ( P `  4
) } ) )  ->  ( ( P `
 0 )  =  ( P `  4
)  ->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  ->  ( P : ( 0 ... 4 ) --> V  -> 
( V USGrph  E  ->  ( ( # `  F
)  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) ) ) )
120119com24 81 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )  /\  ( ( E `  ( F `  2 ) )  =  { ( P `  2 ) ,  ( P ` 
3 ) }  /\  ( E `  ( F `
 3 ) )  =  { ( P `
 3 ) ,  ( P `  4
) } ) )  ->  ( P :
( 0 ... 4
) --> V  ->  (
( F  e. Word  dom  E  /\  Fun  `' F
)  ->  ( ( P `  0 )  =  ( P ` 
4 )  ->  ( V USGrph  E  ->  ( ( # `
 F )  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) ) ) )
12156, 120sylbi 187 . . . . . . . . . . . . . . . . 17  |-  ( A. k  e.  ( 0..^ 4 ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  ->  ( P : ( 0 ... 4 ) --> V  -> 
( ( F  e. Word  dom  E  /\  Fun  `' F )  ->  (
( P `  0
)  =  ( P `
 4 )  -> 
( V USGrph  E  ->  ( ( # `  F
)  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) ) ) )
122121com13 74 . . . . . . . . . . . . . . . 16  |-  ( ( F  e. Word  dom  E  /\  Fun  `' F )  ->  ( P :
( 0 ... 4
) --> V  ->  ( A. k  e.  (
0..^ 4 ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  ->  (
( P `  0
)  =  ( P `
 4 )  -> 
( V USGrph  E  ->  ( ( # `  F
)  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) ) ) )
1231223imp 1145 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e. Word  dom  E  /\  Fun  `' F
)  /\  P :
( 0 ... 4
) --> V  /\  A. k  e.  ( 0..^ 4 ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  ->  (
( P `  0
)  =  ( P `
 4 )  -> 
( V USGrph  E  ->  ( ( # `  F
)  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) )
124123com14 82 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  (
( P `  0
)  =  ( P `
 4 )  -> 
( V USGrph  E  ->  ( ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... 4 ) --> V  /\  A. k  e.  ( 0..^ 4 ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) )
125 fveq2 5541 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  4  ->  ( P `  ( # `  F
) )  =  ( P `  4 ) )
126125eqeq2d 2307 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  (
( P `  0
)  =  ( P `
 ( # `  F
) )  <->  ( P `  0 )  =  ( P `  4
) ) )
127 oveq2 5882 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  4  ->  (
0 ... ( # `  F
) )  =  ( 0 ... 4 ) )
128127feq2d 5396 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  4  ->  ( P : ( 0 ... ( # `  F
) ) --> V  <->  P :
( 0 ... 4
) --> V ) )
129 oveq2 5882 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  4  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 4 ) )
130129raleqdv 2755 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  4  ->  ( A. k  e.  (
0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  A. k  e.  ( 0..^ 4 ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } ) )
131128, 1303anbi23d 1255 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  4  ->  (
( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  <->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... 4 ) --> V  /\  A. k  e.  ( 0..^ 4 ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) ) )
132131imbi1d 308 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  4  ->  (
( ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )  <->  ( ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... 4 ) --> V  /\  A. k  e.  ( 0..^ 4 ) ( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) )
133132imbi2d 307 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  (
( V USGrph  E  ->  ( ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) )  <->  ( V USGrph  E  ->  ( ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... 4 ) --> V  /\  A. k  e.  ( 0..^ 4 ) ( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) )
134124, 126, 1333imtr4d 259 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  4  ->  (
( P `  0
)  =  ( P `
 ( # `  F
) )  ->  ( V USGrph  E  ->  ( (
( F  e. Word  dom  E  /\  Fun  `' F
)  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) )
135134com14 82 . . . . . . . . . . . 12  |-  ( ( ( F  e. Word  dom  E  /\  Fun  `' F
)  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } )  -> 
( ( P ` 
0 )  =  ( P `  ( # `  F ) )  -> 
( V USGrph  E  ->  ( ( # `  F
)  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) )
136135a1d 22 . . . . . . . . . . 11  |-  ( ( ( F  e. Word  dom  E  /\  Fun  `' F
)  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } )  -> 
( ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/)  ->  ( ( P `  0 )  =  ( P `  ( # `  F ) )  ->  ( V USGrph  E  ->  ( ( # `  F )  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) ) )
137136a1d 22 . . . . . . . . . 10  |-  ( ( ( F  e. Word  dom  E  /\  Fun  `' F
)  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) } )  -> 
( Fun  `' ( P  |`  ( 1..^ (
# `  F )
) )  ->  (
( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/)  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  ->  ( V USGrph  E  ->  ( ( # `  F
)  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) ) ) )
1386, 137syl6bi 219 . . . . . . . . 9  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Trails  E ) P  ->  ( Fun  `' ( P  |`  ( 1..^ ( # `  F
) ) )  -> 
( ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/)  ->  ( ( P `  0 )  =  ( P `  ( # `  F ) )  ->  ( V USGrph  E  ->  ( ( # `  F )  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) ) ) ) )
1391383impd 1165 . . . . . . . 8  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( ( F ( V Trails  E
) P  /\  Fun  `' ( P  |`  (
1..^ ( # `  F
) ) )  /\  ( ( P " { 0 ,  (
# `  F ) } )  i^i  ( P " ( 1..^ (
# `  F )
) ) )  =  (/) )  ->  ( ( P `  0 )  =  ( P `  ( # `  F ) )  ->  ( V USGrph  E  ->  ( ( # `  F )  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) ) )
1405, 139sylbid 206 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Paths  E ) P  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  ->  ( V USGrph  E  ->  ( ( # `  F
)  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) ) )
141140imp3a 420 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  ->  ( V USGrph  E  ->  ( ( # `
 F )  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) )
1424, 141sylbid 206 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Cycles  E ) P  ->  ( V USGrph  E  ->  ( ( # `  F
)  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) )
1431423adant1 973 . . . 4  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Cycles  E ) P  ->  ( V USGrph  E  ->  ( ( # `  F
)  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) ) )
1443, 143mpcom 32 . . 3  |-  ( F ( V Cycles  E ) P  ->  ( V USGrph  E  ->  ( ( # `  F )  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) )
145144com12 27 . 2  |-  ( V USGrph  E  ->  ( F ( V Cycles  E ) P  ->  ( ( # `  F )  =  4  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) ) ) )
1461453imp 1145 1  |-  ( ( V USGrph  E  /\  F ( V Cycles  E ) P  /\  ( # `  F
)  =  4 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) )  /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    u. cun 3163    i^i cin 3164   (/)c0 3468   {cpr 3654   class class class wbr 4039   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754    + caddc 8756    <_ cle 8884   2c2 9811   3c3 9812   4c4 9813   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798  ..^cfzo 10886   #chash 11353  Word cword 11419   USGrph cusg 28227   Walks cwalk 28310   Trails ctrail 28311   Paths cpath 28312   Cycles ccycl 28318
This theorem is referenced by:  n4cyclfrgra  28440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-usgra 28229  df-wlk 28319  df-trail 28320  df-pth 28321  df-cycl 28324
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