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Theorem 4cycl4dv4e 23489
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 23453 . . . . 5  |-  ( F ( V Cycles  E ) P  ->  F ( V Walks  E ) P )
2 wlkbprop 23368 . . . . 5  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
31, 2syl 16 . . . 4  |-  ( F ( V Cycles  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
4 iscycl 23446 . . . . . 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 23402 . . . . . . . 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 23371 . . . . . . . . . 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 11612 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0..^ 4 )  =  ( { 0 ,  1 }  u.  { 2 ,  3 } )
87raleqi 2919 . . . . . . . . . . . . . . . . . . 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 3534 . . . . . . . . . . . . . . . . . . 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 10653 . . . . . . . . . . . . . . . . . . . . 21  |-  0  e.  ZZ
11 1z 10672 . . . . . . . . . . . . . . . . . . . . 21  |-  1  e.  ZZ
12 fveq2 5688 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
1312fveq2d 5692 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  0  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  0 )
) )
14 fveq2 5688 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  0  ->  ( P `  k )  =  ( P ` 
0 ) )
15 oveq1 6097 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( k  =  0  ->  (
k  +  1 )  =  ( 0  +  1 ) )
16 0p1e1 10429 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 0  +  1 )  =  1
1715, 16syl6eq 2489 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( k  =  0  ->  (
k  +  1 )  =  1 )
1817fveq2d 5692 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  0  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
1 ) )
1914, 18preq12d 3959 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  0  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
0 ) ,  ( P `  1 ) } )
2013, 19eqeq12d 2455 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  0  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } ) )
21 fveq2 5688 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  1  ->  ( F `  k )  =  ( F ` 
1 ) )
2221fveq2d 5692 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  1  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  1 )
) )
23 fveq2 5688 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  1  ->  ( P `  k )  =  ( P ` 
1 ) )
24 oveq1 6097 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( k  =  1  ->  (
k  +  1 )  =  ( 1  +  1 ) )
25 1p1e2 10431 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 1  +  1 )  =  2
2624, 25syl6eq 2489 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( k  =  1  ->  (
k  +  1 )  =  2 )
2726fveq2d 5692 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  1  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
2 ) )
2823, 27preq12d 3959 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  1  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
1 ) ,  ( P `  2 ) } )
2922, 28eqeq12d 2455 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  1  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } ) )
3020, 29ralprg 3922 . . . . . . . . . . . . . . . . . . . . 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 667 . . . . . . . . . . . . . . . . . . . 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 10674 . . . . . . . . . . . . . . . . . . . . 21  |-  2  e.  ZZ
33 3z 10675 . . . . . . . . . . . . . . . . . . . . 21  |-  3  e.  ZZ
34 fveq2 5688 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  2  ->  ( F `  k )  =  ( F ` 
2 ) )
3534fveq2d 5692 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  2  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  2 )
) )
36 fveq2 5688 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  2  ->  ( P `  k )  =  ( P ` 
2 ) )
37 oveq1 6097 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( k  =  2  ->  (
k  +  1 )  =  ( 2  +  1 ) )
38 2p1e3 10441 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 2  +  1 )  =  3
3937, 38syl6eq 2489 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( k  =  2  ->  (
k  +  1 )  =  3 )
4039fveq2d 5692 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  2  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
3 ) )
4136, 40preq12d 3959 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  2  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
2 ) ,  ( P `  3 ) } )
4235, 41eqeq12d 2455 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  2  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  2
) )  =  {
( P `  2
) ,  ( P `
 3 ) } ) )
43 fveq2 5688 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  3  ->  ( F `  k )  =  ( F ` 
3 ) )
4443fveq2d 5692 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  3  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  3 )
) )
45 fveq2 5688 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  3  ->  ( P `  k )  =  ( P ` 
3 ) )
46 oveq1 6097 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( k  =  3  ->  (
k  +  1 )  =  ( 3  +  1 ) )
47 3p1e4 10443 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 3  +  1 )  =  4
4846, 47syl6eq 2489 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( k  =  3  ->  (
k  +  1 )  =  4 )
4948fveq2d 5692 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( k  =  3  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
4 ) )
5045, 49preq12d 3959 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  3  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
3 ) ,  ( P `  4 ) } )
5144, 50eqeq12d 2455 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  3  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  3
) )  =  {
( P `  3
) ,  ( P `
 4 ) } ) )
5242, 51ralprg 3922 . . . . . . . . . . . . . . . . . . . . 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
) } ) ) )
5332, 33, 52mp2an 667 . . . . . . . . . . . . . . . . . . . 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 ) } ) )
5431, 53anbi12i 692 . . . . . . . . . . . . . . . . . . 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
) } ) ) )
558, 9, 543bitri 271 . . . . . . . . . . . . . . . . . 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
) } ) ) )
56 preq2 3952 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( P `  4 )  =  ( P ` 
0 )  ->  { ( P `  3 ) ,  ( P ` 
4 ) }  =  { ( P ` 
3 ) ,  ( P `  0 ) } )
5756eqcoms 2444 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( P `  0 )  =  ( P ` 
4 )  ->  { ( P `  3 ) ,  ( P ` 
4 ) }  =  { ( P ` 
3 ) ,  ( P `  0 ) } )
5857eqeq2d 2452 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( P `  0 )  =  ( P ` 
4 )  ->  (
( E `  ( F `  3 )
)  =  { ( P `  3 ) ,  ( P ` 
4 ) }  <->  ( E `  ( F `  3
) )  =  {
( P `  3
) ,  ( P `
 0 ) } ) )
5958anbi2d 698 . . . . . . . . . . . . . . . . . . . . . 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
) } ) ) )
6059anbi2d 698 . . . . . . . . . . . . . . . . . . . . 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
) } ) ) ) )
61 simpll 748 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F ) )  /\  ( # `  F )  =  4 )  ->  V USGrph  E )
62 simplrl 754 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F ) )  /\  ( # `  F )  =  4 )  ->  F  e. Word  dom 
E )
63 simplrr 755 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F ) )  /\  ( # `  F )  =  4 )  ->  Fun  `' F
)
64 simpr 458 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F ) )  /\  ( # `  F )  =  4 )  ->  ( # `  F
)  =  4 )
65 4cycl4dv 23488 . . . . . . . . . . . . . . . . . . . . . . . . . . 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
) ) ) ) ) )
6661, 62, 63, 64, 65syl13anc 1215 . . . . . . . . . . . . . . . . . . . . . . . . . 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
) ) ) ) ) )
6766imp 429 . . . . . . . . . . . . . . . . . . . . . . . . 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
) ) ) ) )
68 4nn0 10594 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  4  e.  NN0
6968nn0zi 10667 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  4  e.  ZZ
70 3re 10391 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  3  e.  RR
71 4re 10394 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  4  e.  RR
72 3lt4 10487 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  3  <  4
7370, 71, 72ltleii 9493 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  3  <_  4
74 eluz2 10863 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( 4  e.  ( ZZ>= `  3
)  <->  ( 3  e.  ZZ  /\  4  e.  ZZ  /\  3  <_ 
4 ) )
7533, 69, 73, 74mpbir3an 1165 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  4  e.  ( ZZ>= `  3 )
76 4fvwrd4 11529 . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
7775, 76mpan 665 . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
78 preq12 3953 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( P `  0
)  =  a  /\  ( P `  1 )  =  b )  ->  { ( P ` 
0 ) ,  ( P `  1 ) }  =  { a ,  b } )
7978adantr 462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  { ( P `
 0 ) ,  ( P `  1
) }  =  {
a ,  b } )
8079eleq1d 2507 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
) )
81 simplr 749 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( P ` 
1 )  =  b )
82 simprl 750 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( P ` 
2 )  =  c )
8381, 82preq12d 3959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  { ( P `
 1 ) ,  ( P `  2
) }  =  {
b ,  c } )
8483eleq1d 2507 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
) )
8580, 84anbi12d 705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
86 preq12 3953 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( P `  2
)  =  c  /\  ( P `  3 )  =  d )  ->  { ( P ` 
2 ) ,  ( P `  3 ) }  =  { c ,  d } )
8786adantl 463 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  { ( P `
 2 ) ,  ( P `  3
) }  =  {
c ,  d } )
8887eleq1d 2507 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
) )
89 simprr 751 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( P ` 
3 )  =  d )
90 simpll 748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( P ` 
0 )  =  a )
9189, 90preq12d 3959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  { ( P `
 3 ) ,  ( P `  0
) }  =  {
d ,  a } )
9291eleq1d 2507 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
) )
9388, 92anbi12d 705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
9485, 93anbi12d 705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
9590, 81neeq12d 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( P `
 0 )  =/=  ( P `  1
)  <->  a  =/=  b
) )
9690, 82neeq12d 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( P `
 0 )  =/=  ( P `  2
)  <->  a  =/=  c
) )
9790, 89neeq12d 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( P `
 0 )  =/=  ( P `  3
)  <->  a  =/=  d
) )
9895, 96, 973anbi123d 1284 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
9981, 82neeq12d 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( P `
 1 )  =/=  ( P `  2
)  <->  b  =/=  c
) )
10081, 89neeq12d 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( P `
 1 )  =/=  ( P `  3
)  <->  b  =/=  d
) )
101 simpl 454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( P `  2
)  =  c  /\  ( P `  3 )  =  d )  -> 
( P `  2
)  =  c )
102 simpr 458 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( P `  2
)  =  c  /\  ( P `  3 )  =  d )  -> 
( P `  3
)  =  d )
103101, 102neeq12d 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( P `  2
)  =  c  /\  ( P `  3 )  =  d )  -> 
( ( P ` 
2 )  =/=  ( P `  3 )  <->  c  =/=  d ) )
104103adantl 463 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( P ` 
0 )  =  a  /\  ( P ` 
1 )  =  b )  /\  ( ( P `  2 )  =  c  /\  ( P `  3 )  =  d ) )  ->  ( ( P `
 2 )  =/=  ( P `  3
)  <->  c  =/=  d
) )
10599, 100, 1043anbi123d 1284 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
10698, 105anbi12d 705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
10794, 106anbi12d 705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
108107biimpcd 224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
109108reximdv 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
110109reximdv 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
111110reximdv 2825 . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
112111reximdv 2825 . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
11367, 77, 112syl2im 38 . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
114113exp41 607 . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) ) ) )
115114com14 88 . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) ) ) )
116115com35 90 . . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) ) ) )
11760, 116syl6bi 228 . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) ) ) ) )
118117com12 31 . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) ) ) ) )
119118com24 87 . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) ) ) ) )
12055, 119sylbi 195 . . . . . . . . . . . . . . . . 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 ) ) ) ) ) ) ) ) )
121120com13 80 . . . . . . . . . . . . . . . 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 ) ) ) ) ) ) ) ) )
1221213imp 1176 . . . . . . . . . . . . . . 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 ) ) ) ) ) ) )
123122com14 88 . . . . . . . . . . . . . 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 ) ) ) ) ) ) )
124 fveq2 5688 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  4  ->  ( P `  ( # `  F
) )  =  ( P `  4 ) )
125124eqeq2d 2452 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  4  ->  (
( P `  0
)  =  ( P `
 ( # `  F
) )  <->  ( P `  0 )  =  ( P `  4
) ) )
126 oveq2 6098 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  4  ->  (
0 ... ( # `  F
) )  =  ( 0 ... 4 ) )
127126feq2d 5544 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  4  ->  ( P : ( 0 ... ( # `  F
) ) --> V  <->  P :
( 0 ... 4
) --> V ) )
128 oveq2 6098 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  4  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 4 ) )
129128raleqdv 2921 . . . . . . . . . . . . . . . . 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 ) ) } ) )
130127, 1293anbi23d 1287 . . . . . . . . . . . . . . . 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 ) ) } ) ) )
131130imbi1d 317 . . . . . . . . . . . . . . 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 ) ) ) ) ) )
132131imbi2d 316 . . . . . . . . . . . . . 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 ) ) ) ) ) ) )
133123, 125, 1323imtr4d 268 . . . . . . . . . . . . 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 ) ) ) ) ) ) )
134133com14 88 . . . . . . . . . . . 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 ) ) ) ) ) ) )
135134a1d 25 . . . . . . . . . . 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 ) ) ) ) ) ) ) )
136135a1d 25 . . . . . . . . . 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 ) ) ) ) ) ) ) ) )
1376, 136syl6bi 228 . . . . . . . . 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 ) ) ) ) ) ) ) ) ) )
1381373impd 1196 . . . . . . . 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 ) ) ) ) ) ) ) )
1395, 138sylbid 215 . . . . . . 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 ) ) ) ) ) ) ) )
140139imp3a 431 . . . . . 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 ) ) ) ) ) ) )
1414, 140sylbid 215 . . . . 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 ) ) ) ) ) ) )
1421413adant1 1001 . . . 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 ) ) ) ) ) ) )
1433, 142mpcom 36 . . 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 ) ) ) ) ) )
144143com12 31 . 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 ) ) ) ) ) )
1451443imp 1176 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   _Vcvv 2970    u. cun 3323    i^i cin 3324   (/)c0 3634   {cpr 3876   class class class wbr 4289   `'ccnv 4835   dom cdm 4836   ran crn 4837    |` cres 4838   "cima 4839   Fun wfun 5409   -->wf 5411   ` cfv 5415  (class class class)co 6090   0cc0 9278   1c1 9279    + caddc 9281    <_ cle 9415   2c2 10367   3c3 10368   4c4 10369   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433  ..^cfzo 11544   #chash 12099  Word cword 12217   USGrph cusg 23199   Walks cwalk 23340   Trails ctrail 23341   Paths cpath 23342   Cycles ccycl 23349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-usgra 23201  df-wlk 23350  df-trail 23351  df-pth 23352  df-cycl 23355
This theorem is referenced by:  n4cyclfrgra  30535
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