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Theorem usg2wotspth 30550
Description: A walk of length 2 between two different vertices as ordered triple corresponds to a simple path of length 2 in an undirected simple graph. (Contributed by Alexander van der Vekens, 16-Feb-2018.)
Assertion
Ref Expression
usg2wotspth  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  A  =/=  C
)  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
Distinct variable groups:    A, f, p    B, f, p    C, f, p    f, E, p   
f, V, p

Proof of Theorem usg2wotspth
StepHypRef Expression
1 usgrav 23421 . . . 4  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 el2wlkonotot 30539 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
32expcom 435 . . . . . 6  |-  ( ( A  e.  V  /\  C  e.  V )  ->  ( ( V  e. 
_V  /\  E  e.  _V )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
433adant2 1007 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( V  e. 
_V  /\  E  e.  _V )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
54impcom 430 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
61, 5sylan 471 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
763adant3 1008 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  A  =/=  C
)  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
8 simpr1 994 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  A  =/=  C )  /\  ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )  ->  f ( V Walks 
E ) p )
9 vex 3079 . . . . . . . . . . . . . . . . 17  |-  f  e. 
_V
10 vex 3079 . . . . . . . . . . . . . . . . 17  |-  p  e. 
_V
119, 10pm3.2i 455 . . . . . . . . . . . . . . . 16  |-  ( f  e.  _V  /\  p  e.  _V )
12 is2wlk 23615 . . . . . . . . . . . . . . . 16  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( f  e.  _V  /\  p  e.  _V )
)  ->  ( (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  <->  ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) ) ) )
131, 11, 12sylancl 662 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  ( ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 )  <->  ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) ) ) )
14 usgrafun 23428 . . . . . . . . . . . . . . . . . . . . 21  |-  ( V USGrph  E  ->  Fun  E )
15 c0ex 9490 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  0  e.  _V
1615prid1 4090 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  0  e.  { 0 ,  1 }
17 fzo0to2pr 11730 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 0..^ 2 )  =  {
0 ,  1 }
1816, 17eleqtrri 2541 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  0  e.  ( 0..^ 2 )
19 ffvelrn 5949 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( f : ( 0..^ 2 ) --> dom  E  /\  0  e.  (
0..^ 2 ) )  ->  ( f ` 
0 )  e.  dom  E )
2018, 19mpan2 671 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( f `  0 )  e.  dom  E )
21 fvelrn 5947 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( Fun  E  /\  (
f `  0 )  e.  dom  E )  -> 
( E `  (
f `  0 )
)  e.  ran  E
)
2220, 21sylan2 474 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( E `  (
f `  0 )
)  e.  ran  E
)
23 eleq1 2526 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( E `  ( f `
 0 ) )  =  { ( p `
 0 ) ,  ( p `  1
) }  ->  (
( E `  (
f `  0 )
)  e.  ran  E  <->  { ( p `  0
) ,  ( p `
 1 ) }  e.  ran  E ) )
2422, 23syl5ibcom 220 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  ->  { ( p `
 0 ) ,  ( p `  1
) }  e.  ran  E ) )
25 1ex 9491 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  1  e.  _V
2625prid2 4091 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  1  e.  { 0 ,  1 }
2726, 17eleqtrri 2541 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  1  e.  ( 0..^ 2 )
28 ffvelrn 5949 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( f : ( 0..^ 2 ) --> dom  E  /\  1  e.  (
0..^ 2 ) )  ->  ( f ` 
1 )  e.  dom  E )
2927, 28mpan2 671 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( f `  1 )  e.  dom  E )
30 fvelrn 5947 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( Fun  E  /\  (
f `  1 )  e.  dom  E )  -> 
( E `  (
f `  1 )
)  e.  ran  E
)
3129, 30sylan2 474 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( E `  (
f `  1 )
)  e.  ran  E
)
32 eleq1 2526 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( E `  ( f `
 1 ) )  =  { ( p `
 1 ) ,  ( p `  2
) }  ->  (
( E `  (
f `  1 )
)  e.  ran  E  <->  { ( p `  1
) ,  ( p `
 2 ) }  e.  ran  E ) )
3331, 32syl5ibcom 220 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) }  ->  { ( p `
 1 ) ,  ( p `  2
) }  e.  ran  E ) )
3424, 33anim12d 563 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( Fun  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( ( E `
 ( f ` 
0 ) )  =  { ( p ` 
0 ) ,  ( p `  1 ) }  /\  ( E `
 ( f ` 
1 ) )  =  { ( p ` 
1 ) ,  ( p `  2 ) } )  ->  ( { ( p ` 
0 ) ,  ( p `  1 ) }  e.  ran  E  /\  { ( p ` 
1 ) ,  ( p `  2 ) }  e.  ran  E
) ) )
3514, 34sylan 471 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( ( E `
 ( f ` 
0 ) )  =  { ( p ` 
0 ) ,  ( p `  1 ) }  /\  ( E `
 ( f ` 
1 ) )  =  { ( p ` 
1 ) ,  ( p `  2 ) } )  ->  ( { ( p ` 
0 ) ,  ( p `  1 ) }  e.  ran  E  /\  { ( p ` 
1 ) ,  ( p `  2 ) }  e.  ran  E
) ) )
36 simpllr 758 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( ( V USGrph  E  /\  ( { ( p `  0 ) ,  ( p ` 
1 ) }  e.  ran  E  /\  { ( p `  1 ) ,  ( p ` 
2 ) }  e.  ran  E ) )  /\  p : ( 0 ... 2 ) --> V )  /\  A  =/=  C
)  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  p : ( 0 ... 2 ) --> V )
37 usgraedgrn 23451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( V USGrph  E  /\  { ( p `  0 ) ,  ( p ` 
1 ) }  e.  ran  E )  ->  (
p `  0 )  =/=  ( p `  1
) )
3837ex 434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( V USGrph  E  ->  ( { ( p `  0 ) ,  ( p ` 
1 ) }  e.  ran  E  ->  ( p `  0 )  =/=  ( p `  1
) ) )
39 usgraedgrn 23451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( V USGrph  E  /\  { ( p `  1 ) ,  ( p ` 
2 ) }  e.  ran  E )  ->  (
p `  1 )  =/=  ( p `  2
) )
4039ex 434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( V USGrph  E  ->  ( { ( p `  1 ) ,  ( p ` 
2 ) }  e.  ran  E  ->  ( p `  1 )  =/=  ( p `  2
) ) )
41 simplll 757 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( ( ( p `
 0 )  =/=  ( p `  1
)  /\  ( p `  1 )  =/=  ( p `  2
) )  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  /\  A  =/= 
C )  ->  (
p `  0 )  =/=  ( p `  1
) )
42 simpl 457 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( A  =  ( p `
 0 )  /\  C  =  ( p `  2 ) )  ->  A  =  ( p `  0 ) )
43 simpr 461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( A  =  ( p `
 0 )  /\  C  =  ( p `  2 ) )  ->  C  =  ( p `  2 ) )
4442, 43neeq12d 2730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  |-  ( ( A  =  ( p `
 0 )  /\  C  =  ( p `  2 ) )  ->  ( A  =/= 
C  <->  ( p ` 
0 )  =/=  (
p `  2 )
) )
4544biimpd 207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38  |-  ( ( A  =  ( p `
 0 )  /\  C  =  ( p `  2 ) )  ->  ( A  =/= 
C  ->  ( p `  0 )  =/=  ( p `  2
) ) )
46453adant2 1007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( A  =/= 
C  ->  ( p `  0 )  =/=  ( p `  2
) ) )
4746adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( ( ( ( p ` 
0 )  =/=  (
p `  1 )  /\  ( p `  1
)  =/=  ( p `
 2 ) )  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) )  -> 
( A  =/=  C  ->  ( p `  0
)  =/=  ( p `
 2 ) ) )
4847imp 429 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( ( ( p `
 0 )  =/=  ( p `  1
)  /\  ( p `  1 )  =/=  ( p `  2
) )  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  /\  A  =/= 
C )  ->  (
p `  0 )  =/=  ( p `  2
) )
49 simpllr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( ( ( p `
 0 )  =/=  ( p `  1
)  /\  ( p `  1 )  =/=  ( p `  2
) )  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  /\  A  =/= 
C )  ->  (
p `  1 )  =/=  ( p `  2
) )
5041, 48, 493jca 1168 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( ( p `
 0 )  =/=  ( p `  1
)  /\  ( p `  1 )  =/=  ( p `  2
) )  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  /\  A  =/= 
C )  ->  (
( p `  0
)  =/=  ( p `
 1 )  /\  ( p `  0
)  =/=  ( p `
 2 )  /\  ( p `  1
)  =/=  ( p `
 2 ) ) )
5150exp31 604 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( p `  0
)  =/=  ( p `
 1 )  /\  ( p `  1
)  =/=  ( p `
 2 ) )  ->  ( ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) )  -> 
( A  =/=  C  ->  ( ( p ` 
0 )  =/=  (
p `  1 )  /\  ( p `  0
)  =/=  ( p `
 2 )  /\  ( p `  1
)  =/=  ( p `
 2 ) ) ) ) )
5251a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( V USGrph  E  ->  ( ( ( p `  0 )  =/=  ( p ` 
1 )  /\  (
p `  1 )  =/=  ( p `  2
) )  ->  (
( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( A  =/=  C  ->  ( (
p `  0 )  =/=  ( p `  1
)  /\  ( p `  0 )  =/=  ( p `  2
)  /\  ( p `  1 )  =/=  ( p `  2
) ) ) ) ) )
5338, 40, 52syl2and 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( V USGrph  E  ->  ( ( { ( p `  0
) ,  ( p `
 1 ) }  e.  ran  E  /\  { ( p `  1
) ,  ( p `
 2 ) }  e.  ran  E )  ->  ( ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) )  -> 
( A  =/=  C  ->  ( ( p ` 
0 )  =/=  (
p `  1 )  /\  ( p `  0
)  =/=  ( p `
 2 )  /\  ( p `  1
)  =/=  ( p `
 2 ) ) ) ) ) )
5453imp 429 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( V USGrph  E  /\  ( { ( p ` 
0 ) ,  ( p `  1 ) }  e.  ran  E  /\  { ( p ` 
1 ) ,  ( p `  2 ) }  e.  ran  E
) )  ->  (
( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( A  =/=  C  ->  ( (
p `  0 )  =/=  ( p `  1
)  /\  ( p `  0 )  =/=  ( p `  2
)  /\  ( p `  1 )  =/=  ( p `  2
) ) ) ) )
5554adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( V USGrph  E  /\  ( { ( p ` 
0 ) ,  ( p `  1 ) }  e.  ran  E  /\  { ( p ` 
1 ) ,  ( p `  2 ) }  e.  ran  E
) )  /\  p : ( 0 ... 2 ) --> V )  ->  ( ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) )  -> 
( A  =/=  C  ->  ( ( p ` 
0 )  =/=  (
p `  1 )  /\  ( p `  0
)  =/=  ( p `
 2 )  /\  ( p `  1
)  =/=  ( p `
 2 ) ) ) ) )
5655com23 78 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( V USGrph  E  /\  ( { ( p ` 
0 ) ,  ( p `  1 ) }  e.  ran  E  /\  { ( p ` 
1 ) ,  ( p `  2 ) }  e.  ran  E
) )  /\  p : ( 0 ... 2 ) --> V )  ->  ( A  =/= 
C  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( p `
 0 )  =/=  ( p `  1
)  /\  ( p `  0 )  =/=  ( p `  2
)  /\  ( p `  1 )  =/=  ( p `  2
) ) ) ) )
5756imp 429 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( ( V USGrph  E  /\  ( { ( p `
 0 ) ,  ( p `  1
) }  e.  ran  E  /\  { ( p `
 1 ) ,  ( p `  2
) }  e.  ran  E ) )  /\  p : ( 0 ... 2 ) --> V )  /\  A  =/=  C
)  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( p `
 0 )  =/=  ( p `  1
)  /\  ( p `  0 )  =/=  ( p `  2
)  /\  ( p `  1 )  =/=  ( p `  2
) ) ) )
5857imp 429 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( ( V USGrph  E  /\  ( { ( p `  0 ) ,  ( p ` 
1 ) }  e.  ran  E  /\  { ( p `  1 ) ,  ( p ` 
2 ) }  e.  ran  E ) )  /\  p : ( 0 ... 2 ) --> V )  /\  A  =/=  C
)  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( ( p `
 0 )  =/=  ( p `  1
)  /\  ( p `  0 )  =/=  ( p `  2
)  /\  ( p `  1 )  =/=  ( p `  2
) ) )
59 eqid 2454 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( 0 ... 2 )  =  ( 0 ... 2
)
6059f13idfv 30295 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( p : ( 0 ... 2 ) -1-1-> V  <->  ( p : ( 0 ... 2 ) --> V  /\  ( ( p ` 
0 )  =/=  (
p `  1 )  /\  ( p `  0
)  =/=  ( p `
 2 )  /\  ( p `  1
)  =/=  ( p `
 2 ) ) ) )
6136, 58, 60sylanbrc 664 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( ( V USGrph  E  /\  ( { ( p `  0 ) ,  ( p ` 
1 ) }  e.  ran  E  /\  { ( p `  1 ) ,  ( p ` 
2 ) }  e.  ran  E ) )  /\  p : ( 0 ... 2 ) --> V )  /\  A  =/=  C
)  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  p : ( 0 ... 2 )
-1-1-> V )
62 df-f1 5530 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p : ( 0 ... 2 ) -1-1-> V  <->  ( p : ( 0 ... 2 ) --> V  /\  Fun  `' p ) )
6361, 62sylib 196 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( ( V USGrph  E  /\  ( { ( p `  0 ) ,  ( p ` 
1 ) }  e.  ran  E  /\  { ( p `  1 ) ,  ( p ` 
2 ) }  e.  ran  E ) )  /\  p : ( 0 ... 2 ) --> V )  /\  A  =/=  C
)  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( p : ( 0 ... 2
) --> V  /\  Fun  `' p ) )
6463simprd 463 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( ( V USGrph  E  /\  ( { ( p `  0 ) ,  ( p ` 
1 ) }  e.  ran  E  /\  { ( p `  1 ) ,  ( p ` 
2 ) }  e.  ran  E ) )  /\  p : ( 0 ... 2 ) --> V )  /\  A  =/=  C
)  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  Fun  `' p
)
6564exp31 604 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( V USGrph  E  /\  ( { ( p ` 
0 ) ,  ( p `  1 ) }  e.  ran  E  /\  { ( p ` 
1 ) ,  ( p `  2 ) }  e.  ran  E
) )  /\  p : ( 0 ... 2 ) --> V )  ->  ( A  =/= 
C  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  Fun  `' p
) ) )
6665exp31 604 . . . . . . . . . . . . . . . . . . . . 21  |-  ( V USGrph  E  ->  ( ( { ( p `  0
) ,  ( p `
 1 ) }  e.  ran  E  /\  { ( p `  1
) ,  ( p `
 2 ) }  e.  ran  E )  ->  ( p : ( 0 ... 2
) --> V  ->  ( A  =/=  C  ->  (
( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  Fun  `' p
) ) ) ) )
6766adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( { ( p `  0 ) ,  ( p ` 
1 ) }  e.  ran  E  /\  { ( p `  1 ) ,  ( p ` 
2 ) }  e.  ran  E )  ->  (
p : ( 0 ... 2 ) --> V  ->  ( A  =/= 
C  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  Fun  `' p
) ) ) ) )
6835, 67syld 44 . . . . . . . . . . . . . . . . . . 19  |-  ( ( V USGrph  E  /\  f : ( 0..^ 2 ) --> dom  E )  ->  ( ( ( E `
 ( f ` 
0 ) )  =  { ( p ` 
0 ) ,  ( p `  1 ) }  /\  ( E `
 ( f ` 
1 ) )  =  { ( p ` 
1 ) ,  ( p `  2 ) } )  ->  (
p : ( 0 ... 2 ) --> V  ->  ( A  =/= 
C  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  Fun  `' p
) ) ) ) )
6968expcom 435 . . . . . . . . . . . . . . . . . 18  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( V USGrph  E  ->  ( ( ( E `  (
f `  0 )
)  =  { ( p `  0 ) ,  ( p ` 
1 ) }  /\  ( E `  ( f `
 1 ) )  =  { ( p `
 1 ) ,  ( p `  2
) } )  -> 
( p : ( 0 ... 2 ) --> V  ->  ( A  =/=  C  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  Fun  `' p
) ) ) ) ) )
7069com24 87 . . . . . . . . . . . . . . . . 17  |-  ( f : ( 0..^ 2 ) --> dom  E  ->  ( p : ( 0 ... 2 ) --> V  ->  ( ( ( E `  ( f `
 0 ) )  =  { ( p `
 0 ) ,  ( p `  1
) }  /\  ( E `  ( f `  1 ) )  =  { ( p `
 1 ) ,  ( p `  2
) } )  -> 
( V USGrph  E  ->  ( A  =/=  C  -> 
( ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) )  ->  Fun  `' p ) ) ) ) ) )
71703imp 1182 . . . . . . . . . . . . . . . 16  |-  ( ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `
 ( f ` 
0 ) )  =  { ( p ` 
0 ) ,  ( p `  1 ) }  /\  ( E `
 ( f ` 
1 ) )  =  { ( p ` 
1 ) ,  ( p `  2 ) } ) )  -> 
( V USGrph  E  ->  ( A  =/=  C  -> 
( ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) )  ->  Fun  `' p ) ) ) )
7271com12 31 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  ( ( f : ( 0..^ 2 ) --> dom  E  /\  p : ( 0 ... 2 ) --> V  /\  ( ( E `  ( f `  0
) )  =  {
( p `  0
) ,  ( p `
 1 ) }  /\  ( E `  ( f `  1
) )  =  {
( p `  1
) ,  ( p `
 2 ) } ) )  ->  ( A  =/=  C  ->  (
( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  Fun  `' p
) ) ) )
7313, 72sylbid 215 . . . . . . . . . . . . . 14  |-  ( V USGrph  E  ->  ( ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 )  ->  ( A  =/=  C  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  Fun  `' p
) ) ) )
7473com14 88 . . . . . . . . . . . . 13  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 )  ->  ( A  =/=  C  ->  ( V USGrph  E  ->  Fun  `' p
) ) ) )
7574com12 31 . . . . . . . . . . . 12  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  (
( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( A  =/=  C  ->  ( V USGrph  E  ->  Fun  `' p
) ) ) )
76753impia 1185 . . . . . . . . . . 11  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( A  =/= 
C  ->  ( V USGrph  E  ->  Fun  `' p
) ) )
7776com13 80 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( A  =/= 
C  ->  ( (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  Fun  `' p
) ) )
7877a1d 25 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( A  =/=  C  ->  (
( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  ->  Fun  `' p
) ) ) )
79783imp 1182 . . . . . . . 8  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  A  =/=  C
)  ->  ( (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  Fun  `' p
) )
8079imp 429 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  A  =/=  C )  /\  ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )  ->  Fun  `' p
)
81 wlkdvspth 23658 . . . . . . 7  |-  ( ( f ( V Walks  E
) p  /\  Fun  `' p )  ->  f
( V SPaths  E )
p )
828, 80, 81syl2anc 661 . . . . . 6  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  A  =/=  C )  /\  ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )  ->  f ( V SPaths  E ) p )
83 simpr2 995 . . . . . 6  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  A  =/=  C )  /\  ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )  ->  ( # `  f
)  =  2 )
84 simpr3 996 . . . . . 6  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  A  =/=  C )  /\  ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )  ->  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) )
8582, 83, 843jca 1168 . . . . 5  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  A  =/=  C )  /\  ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )  ->  ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )
8685ex 434 . . . 4  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  A  =/=  C
)  ->  ( (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
87 spthispth 23623 . . . . . 6  |-  ( f ( V SPaths  E ) p  ->  f ( V Paths  E ) p )
88 pthistrl 23622 . . . . . 6  |-  ( f ( V Paths  E ) p  ->  f ( V Trails  E ) p )
89 trliswlk 23589 . . . . . 6  |-  ( f ( V Trails  E ) p  ->  f ( V Walks  E ) p )
9087, 88, 893syl 20 . . . . 5  |-  ( f ( V SPaths  E ) p  ->  f ( V Walks  E ) p )
91903anim1i 1174 . . . 4  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )
9286, 91impbid1 203 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  A  =/=  C
)  ->  ( (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  <-> 
( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
93922exbidv 1683 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  A  =/=  C
)  ->  ( E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) )  <->  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
947, 93bitrd 253 1  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  A  =/=  C
)  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2647   _Vcvv 3076   {cpr 3986   <.cotp 3992   class class class wbr 4399   `'ccnv 4946   dom cdm 4947   ran crn 4948   Fun wfun 5519   -->wf 5521   -1-1->wf1 5522   ` cfv 5525  (class class class)co 6199   0cc0 9392   1c1 9393   2c2 10481   ...cfz 11553  ..^cfzo 11664   #chash 12219   USGrph cusg 23415   Walks cwalk 23556   Trails ctrail 23557   Paths cpath 23558   SPaths cspath 23559   2WalksOnOt c2wlkonot 30521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-ot 3993  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-3 10491  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-fzo 11665  df-hash 12220  df-word 12346  df-usgra 23417  df-wlk 23566  df-trail 23567  df-pth 23568  df-spth 23569  df-wlkon 23572  df-2wlkonot 30524
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator