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Theorem usgra2wlkspth 28370
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 21563 . . . 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 733 . . . . . 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 21558 . . . . . . . . . . . . . . . 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 978 . . . . . . . . . . . . . . 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 21 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F  e. Word  dom  E  ->  F  e. Word  dom  E )
653ad2ant1 979 . . . . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . . . 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 21414 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> ran  E )
983ad2ant1 979 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  E : dom  E -1-1-> ran  E
)
109adantl 454 . . . . . . . . . . . . . . . . . . . . 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 959 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  ( # `
 F )  =  2 )
1211adantl 454 . . . . . . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B )  ->  ( P `  0 )  =  A )
1514ad3antlr 713 . . . . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B )  ->  ( P `  ( # `  F
) )  =  B )
1716ad3antlr 713 . . . . . . . . . . . . . . . . . . . . . 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 960 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  A  =/=  B )
1918adantl 454 . . . . . . . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . . . . . . . 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 960 . . . . . . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . . . . . 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 520 . . . . . . . . . . . . . . . . . . . 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 28368 . . . . . . . . . . . . . . . . . . . 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 520 . . . . . . . . . . . . . . . . . 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 959 . . . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . . . 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 589 . . . . . . . . . . . . . . . 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 589 . . . . . . . . . . . . . . 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 208 . . . . . . . . . . . . . 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 77 . . . . . . . . . . . . 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 425 . . . . . . . . . . . 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 76 . . . . . . . . . . 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 1148 . . . . . . . . . 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 421 . . . . . . . . 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 423 . . . . . . . 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 21 . . . . . . . . . . 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 978 . . . . . . . . . 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 712 . . . . . . . . 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 21568 . . . . . . . . 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 225 . . . . . . 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 21559 . . . . . . . . . . . . 13  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
46 simpl 445 . . . . . . . . . . . . 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 979 . . . . . . . . . . 11  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  F  e. Word  dom 
E )
4948ad3antlr 713 . . . . . . . . . 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 454 . . . . . . . . . 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 520 . . . . . . . . 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 21 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  V USGrph  E )
53523ad2ant1 979 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  V USGrph  E )
5453adantl 454 . . . . . . . . . 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 449 . . . . . . . . . . . . 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 979 . . . . . . . . . . 11  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  P :
( 0 ... ( # `
 F ) ) --> V )
5857ad3antlr 713 . . . . . . . . . 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 520 . . . . . . . . 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 520 . . . . . . . 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 959 . . . . . . . . . . 11  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  ( P `  0 )  =  A )
6261ad3antlr 713 . . . . . . . . . 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 960 . . . . . . . . . . 11  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  ( P `  ( # `  F
) )  =  B )
6463ad3antlr 713 . . . . . . . . . 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 454 . . . . . . . . . 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 1135 . . . . . . . . 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 221 . . . . . . . . . . . . 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 30 . . . . . . . . . . . 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 979 . . . . . . . . . . 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 421 . . . . . . . . . 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 708 . . . . . . . . 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 520 . . . . . . . 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 28369 . . . . . . . 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 21600 . . . . . . . 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 890 . . . . . 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 21614 . . . . . . 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 712 . . . . . 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 890 . . . . 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 589 . . . 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 35 . . 3  |-  ( F ( A ( V WalkOn  E ) B ) P  ->  ( ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B
)  ->  F ( A ( V SPathOn  E
) B ) P ) )
8382com12 30 . 2  |-  ( ( V USGrph  E  /\  ( # `
 F )  =  2  /\  A  =/= 
B )  ->  ( F ( A ( V WalkOn  E ) B ) P  ->  F
( A ( V SPathOn  E ) B ) P ) )
84 spthonprp 21616 . . 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 445 . . . 4  |-  ( ( F ( A ( V WalkOn  E ) B ) P  /\  F
( V SPaths  E ) P )  ->  F
( A ( V WalkOn  E ) B ) P )
8685adantl 454 . . 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 196 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605   A.wral 2711   _Vcvv 2962   {cpr 3839   class class class wbr 4237   `'ccnv 4906   dom cdm 4907   ran crn 4908   Fun wfun 5477   -->wf 5479   -1-1->wf1 5480   ` cfv 5483  (class class class)co 6110   0cc0 9021   1c1 9022    + caddc 9024   2c2 10080   ...cfz 11074  ..^cfzo 11166   #chash 11649  Word cword 11748   USGrph cusg 21396   Walks cwalk 21537   Trails ctrail 21538   SPaths cspath 21540   WalkOn cwlkon 21541   SPathOn cspthon 21544
This theorem is referenced by:  2pthwlkonot  28441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-map 7049  df-pm 7050  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-card 7857  df-cda 8079  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-n0 10253  df-z 10314  df-uz 10520  df-fz 11075  df-fzo 11167  df-hash 11650  df-word 11754  df-usgra 21398  df-wlk 21547  df-trail 21548  df-pth 21549  df-spth 21550  df-wlkon 21553  df-spthon 21556
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