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Theorem usgra2wlkspth 24285
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 24199 . . . 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 24184 . . . . . . . . . . . . . . . 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 1011 . . . . . . . . . . . . . . 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 1012 . . . . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . . . 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 24025 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> ran  E )
983ad2ant1 1012 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  E : dom  E -1-1-> ran  E
)
109adantl 466 . . . . . . . . . . . . . . . . . . . . 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 992 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  ( # `
 F )  =  2 )
1211adantl 466 . . . . . . . . . . . . . . . . . . . . 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 1171 . . . . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B )  ->  ( P `  0 )  =  A )
1514ad3antlr 730 . . . . . . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B )  ->  ( P `  ( # `  F
) )  =  B )
1716ad3antlr 730 . . . . . . . . . . . . . . . . . . . . . 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 993 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  A  =/=  B )
1918adantl 466 . . . . . . . . . . . . . . . . . . . . . 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 1171 . . . . . . . . . . . . . . . . . . . . 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 993 . . . . . . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . . . . . 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 532 . . . . . . . . . . . . . . . . . . . 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 24283 . . . . . . . . . . . . . . . . . . . 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 60 . . . . . . . . . . . . . . . . . . 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 532 . . . . . . . . . . . . . . . . . 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 992 . . . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . . 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 1171 . . . . . . . . . . . . . . . . 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 604 . . . . . . . . . . . . . . . 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 604 . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . 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 1185 . . . . . . . . . 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 430 . . . . . . . . 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 432 . . . . . . . 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 1011 . . . . . . . . . 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 729 . . . . . . . . 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 24203 . . . . . . . . 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 16 . . . . . . . 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 24185 . . . . . . . . . . . . 13  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
46 simpl 457 . . . . . . . . . . . . 13  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  F  e. Word  dom  E )
4745, 46syl 16 . . . . . . . . . . . 12  |-  ( F ( V Walks  E ) P  ->  F  e. Word  dom 
E )
48473ad2ant1 1012 . . . . . . . . . . 11  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  F  e. Word  dom 
E )
4948ad3antlr 730 . . . . . . . . . 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 466 . . . . . . . . . 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 532 . . . . . . . . 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 1012 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  V USGrph  E )
5453adantl 466 . . . . . . . . . 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 461 . . . . . . . . . . . . 13  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  P : ( 0 ... ( # `  F ) ) --> V )
5645, 55syl 16 . . . . . . . . . . . 12  |-  ( F ( V Walks  E ) P  ->  P :
( 0 ... ( # `
 F ) ) --> V )
57563ad2ant1 1012 . . . . . . . . . . 11  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  P :
( 0 ... ( # `
 F ) ) --> V )
5857ad3antlr 730 . . . . . . . . . 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 532 . . . . . . . . 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 532 . . . . . . . 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 992 . . . . . . . . . . 11  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  ( P `  0 )  =  A )
6261ad3antlr 730 . . . . . . . . . 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 993 . . . . . . . . . . 11  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  ( P `  ( # `  F
) )  =  B )
6463ad3antlr 730 . . . . . . . . . 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 466 . . . . . . . . . 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 1171 . . . . . . . . 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 31 . . . . . . . . . . . 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 1012 . . . . . . . . . . 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 430 . . . . . . . . . 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 725 . . . . . . . . 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 532 . . . . . . . 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 24284 . . . . . . . 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 60 . . . . . . 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 24235 . . . . . . . 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 16 . . . . . . 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 915 . . . . . 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 24249 . . . . . . 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 729 . . . . . 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 915 . . . . 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 604 . . . 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 36 . . 3  |-  ( F ( A ( V WalkOn  E ) B ) P  ->  ( ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
)  ->  F ( A ( V SPathOn  E
) B ) P ) )
8382com12 31 . 2  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  ( F ( A ( V WalkOn  E ) B ) P  ->  F
( A ( V SPathOn  E ) B ) P ) )
84 spthonprp 24251 . . 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 457 . . . 4  |-  ( ( F ( A ( V WalkOn  E ) B ) P  /\  F
( V SPaths  E ) P )  ->  F
( A ( V WalkOn  E ) B ) P )
8685adantl 466 . . 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 16 . 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   A.wral 2809   _Vcvv 3108   {cpr 4024   class class class wbr 4442   `'ccnv 4993   dom cdm 4994   ran crn 4995   Fun wfun 5575   -->wf 5577   -1-1->wf1 5578   ` cfv 5581  (class class class)co 6277   0cc0 9483   1c1 9484    + caddc 9486   2c2 10576   ...cfz 11663  ..^cfzo 11783   #chash 12362  Word cword 12489   USGrph cusg 23995   Walks cwalk 24162   Trails ctrail 24163   SPaths cspath 24165   WalkOn cwlkon 24166   SPathOn cspthon 24169
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-fzo 11784  df-hash 12363  df-word 12497  df-usgra 23998  df-wlk 24172  df-trail 24173  df-pth 24174  df-spth 24175  df-wlkon 24178  df-spthon 24181
This theorem is referenced by:  2pthwlkonot  24549
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