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Theorem usgra2wlkspth 25025
Description: In a undirected simple graph, any walk of length 2 between two different vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
Assertion
Ref Expression
usgra2wlkspth  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  ( F ( A ( V WalkOn  E ) B ) P  <->  F ( A ( V SPathOn  E
) B ) P ) )

Proof of Theorem usgra2wlkspth
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 wlkonprop 24939 . . . 4  |-  ( F ( A ( V WalkOn  E ) B ) P  ->  ( (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) ) )
2 simplr 754 . . . . . 6  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  F
( A ( V WalkOn  E ) B ) P )
3 iswlk 24924 . . . . . . . . . . . . . . . 16  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Walks  E ) P 
<->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) ) )
433adant3 1017 . . . . . . . . . . . . . . 15  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( F ( V Walks  E
) P  <->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) ) )
5 id 22 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F  e. Word  dom  E  ->  F  e. Word  dom  E )
653ad2ant1 1018 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  F  e. Word  dom  E )
76ad4antlr 731 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  /\  ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  /\  F ( A ( V WalkOn  E
) B ) P )  /\  ( V USGrph  E  /\  ( # `  F
)  =  2  /\  A  =/=  B ) )  ->  F  e. Word  dom 
E )
8 usgraf1 24764 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> ran  E )
983ad2ant1 1018 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  E : dom  E -1-1-> ran  E
)
109adantl 464 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  /\  ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  /\  F ( A ( V WalkOn  E
) B ) P )  /\  ( V USGrph  E  /\  ( # `  F
)  =  2  /\  A  =/=  B ) )  ->  E : dom  E -1-1-> ran  E )
11 simp2 998 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  ( # `
 F )  =  2 )
1211adantl 464 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  /\  ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  /\  F ( A ( V WalkOn  E
) B ) P )  /\  ( V USGrph  E  /\  ( # `  F
)  =  2  /\  A  =/=  B ) )  ->  ( # `  F
)  =  2 )
137, 10, 123jca 1177 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  /\  ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  /\  F ( A ( V WalkOn  E
) B ) P )  /\  ( V USGrph  E  /\  ( # `  F
)  =  2  /\  A  =/=  B ) )  ->  ( F  e. Word  dom  E  /\  E : dom  E -1-1-> ran  E  /\  ( # `  F
)  =  2 ) )
14 simpl 455 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B )  ->  ( P `  0 )  =  A )
1514ad3antlr 729 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  /\  ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  /\  F ( A ( V WalkOn  E
) B ) P )  /\  ( V USGrph  E  /\  ( # `  F
)  =  2  /\  A  =/=  B ) )  ->  ( P `  0 )  =  A )
16 simpr 459 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B )  ->  ( P `  ( # `  F
) )  =  B )
1716ad3antlr 729 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  /\  ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  /\  F ( A ( V WalkOn  E
) B ) P )  /\  ( V USGrph  E  /\  ( # `  F
)  =  2  /\  A  =/=  B ) )  ->  ( P `  ( # `  F
) )  =  B )
18 simp3 999 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  A  =/=  B )
1918adantl 464 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  /\  ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  /\  F ( A ( V WalkOn  E
) B ) P )  /\  ( V USGrph  E  /\  ( # `  F
)  =  2  /\  A  =/=  B ) )  ->  A  =/=  B )
2015, 17, 193jca 1177 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  /\  ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  /\  F ( A ( V WalkOn  E
) B ) P )  /\  ( V USGrph  E  /\  ( # `  F
)  =  2  /\  A  =/=  B ) )  ->  ( ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B  /\  A  =/=  B
) )
21 simp3 999 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )
2221ad4antlr 731 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  /\  ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  /\  F ( A ( V WalkOn  E
) B ) P )  /\  ( V USGrph  E  /\  ( # `  F
)  =  2  /\  A  =/=  B ) )  ->  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )
2320, 22jca 530 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  /\  ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  /\  F ( A ( V WalkOn  E
) B ) P )  /\  ( V USGrph  E  /\  ( # `  F
)  =  2  /\  A  =/=  B ) )  ->  ( (
( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )
24 usgra2wlkspthlem1 25023 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F  e. Word  dom  E  /\  E : dom  E -1-1-> ran 
E  /\  ( # `  F
)  =  2 )  ->  ( ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  Fun  `' F ) )
2513, 23, 24sylc 59 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  /\  ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  /\  F ( A ( V WalkOn  E
) B ) P )  /\  ( V USGrph  E  /\  ( # `  F
)  =  2  /\  A  =/=  B ) )  ->  Fun  `' F
)
267, 25jca 530 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  /\  ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  /\  F ( A ( V WalkOn  E
) B ) P )  /\  ( V USGrph  E  /\  ( # `  F
)  =  2  /\  A  =/=  B ) )  ->  ( F  e. Word  dom  E  /\  Fun  `' F ) )
27 simp2 998 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  P : ( 0 ... ( # `  F
) ) --> V )
2827ad4antlr 731 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  /\  ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  /\  F ( A ( V WalkOn  E
) B ) P )  /\  ( V USGrph  E  /\  ( # `  F
)  =  2  /\  A  =/=  B ) )  ->  P :
( 0 ... ( # `
 F ) ) --> V )
2926, 28, 223jca 1177 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  /\  ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  /\  F ( A ( V WalkOn  E
) B ) P )  /\  ( V USGrph  E  /\  ( # `  F
)  =  2  /\  A  =/=  B ) )  ->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )
3029exp31 602 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )
)  /\  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  /\  ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  ->  ( F
( A ( V WalkOn  E ) B ) P  ->  ( ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
)  ->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) ) ) )
3130exp31 602 . . . . . . . . . . . . . . 15  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  (
( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  -> 
( ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  B )  ->  ( F
( A ( V WalkOn  E ) B ) P  ->  ( ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
)  ->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) ) ) ) ) )
324, 31sylbid 215 . . . . . . . . . . . . . 14  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( F ( V Walks  E
) P  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B )  ->  ( F ( A ( V WalkOn  E
) B ) P  ->  ( ( V USGrph  E  /\  ( # `  F
)  =  2  /\  A  =/=  B )  ->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) ) ) ) ) )
3332com13 80 . . . . . . . . . . . . 13  |-  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B )  ->  ( F ( V Walks  E
) P  ->  (
( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )
)  ->  ( F
( A ( V WalkOn  E ) B ) P  ->  ( ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
)  ->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) ) ) ) ) )
3433ex 432 . . . . . . . . . . . 12  |-  ( ( P `  0 )  =  A  ->  (
( P `  ( # `
 F ) )  =  B  ->  ( F ( V Walks  E
) P  ->  (
( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )
)  ->  ( F
( A ( V WalkOn  E ) B ) P  ->  ( ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
)  ->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) ) ) ) ) ) )
3534com3r 79 . . . . . . . . . . 11  |-  ( F ( V Walks  E ) P  ->  ( ( P `  0 )  =  A  ->  ( ( P `  ( # `  F ) )  =  B  ->  ( (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( F ( A ( V WalkOn  E ) B ) P  ->  (
( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B )  ->  (
( F  e. Word  dom  E  /\  Fun  `' F
)  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) ) ) ) ) ) )
36353imp 1191 . . . . . . . . . 10  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  ( (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( F ( A ( V WalkOn  E ) B ) P  ->  (
( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B )  ->  (
( F  e. Word  dom  E  /\  Fun  `' F
)  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) ) ) ) )
3736impcom 428 . . . . . . . . 9  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )
)  /\  ( F
( V Walks  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  ->  ( F
( A ( V WalkOn  E ) B ) P  ->  ( ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
)  ->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) ) ) )
3837imp31 430 . . . . . . . 8  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  (
( F  e. Word  dom  E  /\  Fun  `' F
)  /\  P :
( 0 ... ( # `
 F ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) )
39 id 22 . . . . . . . . . . 11  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
40393adant3 1017 . . . . . . . . . 10  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )
4140ad3antrrr 728 . . . . . . . . 9  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )
42 istrl 24943 . . . . . . . . 9  |-  ( ( ( 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. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) ) )
4341, 42syl 17 . . . . . . . 8  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  ( F ( V Trails  E
) P  <->  ( ( F  e. Word  dom  E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F
) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) ) )
4438, 43mpbird 232 . . . . . . 7  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  F
( V Trails  E ) P )
45 2mwlk 24925 . . . . . . . . . . . . 13  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
46 simpl 455 . . . . . . . . . . . . 13  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  F  e. Word  dom  E )
4745, 46syl 17 . . . . . . . . . . . 12  |-  ( F ( V Walks  E ) P  ->  F  e. Word  dom 
E )
48473ad2ant1 1018 . . . . . . . . . . 11  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  F  e. Word  dom 
E )
4948ad3antlr 729 . . . . . . . . . 10  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  F  e. Word  dom  E )
5011adantl 464 . . . . . . . . . 10  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  ( # `
 F )  =  2 )
5149, 50jca 530 . . . . . . . . 9  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  ( F  e. Word  dom  E  /\  ( # `  F )  =  2 ) )
52 id 22 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  V USGrph  E )
53523ad2ant1 1018 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  V USGrph  E )
5453adantl 464 . . . . . . . . . 10  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  V USGrph  E )
55 simpr 459 . . . . . . . . . . . . 13  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  P : ( 0 ... ( # `  F ) ) --> V )
5645, 55syl 17 . . . . . . . . . . . 12  |-  ( F ( V Walks  E ) P  ->  P :
( 0 ... ( # `
 F ) ) --> V )
57563ad2ant1 1018 . . . . . . . . . . 11  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  P :
( 0 ... ( # `
 F ) ) --> V )
5857ad3antlr 729 . . . . . . . . . 10  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  P : ( 0 ... ( # `  F
) ) --> V )
5954, 58jca 530 . . . . . . . . 9  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  ( V USGrph  E  /\  P :
( 0 ... ( # `
 F ) ) --> V ) )
6051, 59jca 530 . . . . . . . 8  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  (
( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  ( V USGrph  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) ) )
61 simp2 998 . . . . . . . . . . 11  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  ( P `  0 )  =  A )
6261ad3antlr 729 . . . . . . . . . 10  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  ( P `  0 )  =  A )
63 simp3 999 . . . . . . . . . . 11  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  ( P `  ( # `  F
) )  =  B )
6463ad3antlr 729 . . . . . . . . . 10  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  ( P `  ( # `  F
) )  =  B )
6518adantl 464 . . . . . . . . . 10  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  A  =/=  B )
6662, 64, 653jca 1177 . . . . . . . . 9  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  (
( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B ) )
674, 21syl6bi 228 . . . . . . . . . . . . 13  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( F ( V Walks  E
) P  ->  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) )
6867com12 29 . . . . . . . . . . . 12  |-  ( F ( V Walks  E ) P  ->  ( (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) )
69683ad2ant1 1018 . . . . . . . . . . 11  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  ( (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } ) )
7069impcom 428 . . . . . . . . . 10  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )
)  /\  ( F
( V Walks  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  ->  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )
7170ad2antrr 724 . . . . . . . . 9  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )
7266, 71jca 530 . . . . . . . 8  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )
73 usgra2wlkspthlem2 25024 . . . . . . . 8  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  ( V USGrph  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )  ->  ( (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  Fun  `' P ) )
7460, 72, 73sylc 59 . . . . . . 7  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  Fun  `' P )
75 isspth 24975 . . . . . . . 8  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V SPaths  E ) P 
<->  ( F ( V Trails  E ) P  /\  Fun  `' P ) ) )
7641, 75syl 17 . . . . . . 7  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  ( F ( V SPaths  E
) P  <->  ( F
( V Trails  E ) P  /\  Fun  `' P
) ) )
7744, 74, 76mpbir2and 923 . . . . . 6  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  F
( V SPaths  E ) P )
78 isspthon 24989 . . . . . . 7  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( F ( A ( V SPathOn  E ) B ) P  <->  ( F ( A ( V WalkOn  E
) B ) P  /\  F ( V SPaths  E ) P ) ) )
7978ad3antrrr 728 . . . . . 6  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  ( F ( A ( V SPathOn  E ) B ) P  <->  ( F ( A ( V WalkOn  E
) B ) P  /\  F ( V SPaths  E ) P ) ) )
802, 77, 79mpbir2and 923 . . . . 5  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B ) )  /\  F
( A ( V WalkOn  E ) B ) P )  /\  ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
) )  ->  F
( A ( V SPathOn  E ) B ) P )
8180exp31 602 . . . 4  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )
)  /\  ( F
( V Walks  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) )  ->  ( F
( A ( V WalkOn  E ) B ) P  ->  ( ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
)  ->  F ( A ( V SPathOn  E
) B ) P ) ) )
821, 81mpcom 34 . . 3  |-  ( F ( A ( V WalkOn  E ) B ) P  ->  ( ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
)  ->  F ( A ( V SPathOn  E
) B ) P ) )
8382com12 29 . 2  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  ( F ( A ( V WalkOn  E ) B ) P  ->  F
( A ( V SPathOn  E ) B ) P ) )
84 spthonprp 24991 . . 3  |-  ( F ( A ( V SPathOn  E ) B ) P  ->  ( (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( A ( V WalkOn  E ) B ) P  /\  F
( V SPaths  E ) P ) ) )
85 simpl 455 . . . 4  |-  ( ( F ( A ( V WalkOn  E ) B ) P  /\  F
( V SPaths  E ) P )  ->  F
( A ( V WalkOn  E ) B ) P )
8685adantl 464 . . 3  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )
)  /\  ( F
( A ( V WalkOn  E ) B ) P  /\  F ( V SPaths  E ) P ) )  ->  F
( A ( V WalkOn  E ) B ) P )
8784, 86syl 17 . 2  |-  ( F ( A ( V SPathOn  E ) B ) P  ->  F ( A ( V WalkOn  E
) B ) P )
8883, 87impbid1 203 1  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  ( F ( A ( V WalkOn  E ) B ) P  <->  F ( A ( V SPathOn  E
) B ) P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   _Vcvv 3058   {cpr 3973   class class class wbr 4394   `'ccnv 4821   dom cdm 4822   ran crn 4823   Fun wfun 5562   -->wf 5564   -1-1->wf1 5565   ` cfv 5568  (class class class)co 6277   0cc0 9521   1c1 9522    + caddc 9524   2c2 10625   ...cfz 11724  ..^cfzo 11852   #chash 12450  Word cword 12581   USGrph cusg 24734   Walks cwalk 24902   Trails ctrail 24903   SPaths cspath 24905   WalkOn cwlkon 24906   SPathOn cspthon 24909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589  df-usgra 24737  df-wlk 24912  df-trail 24913  df-pth 24914  df-spth 24915  df-wlkon 24918  df-spthon 24921
This theorem is referenced by:  2pthwlkonot  25289
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