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Theorem disjxwwlks 24440
Description: Sets of walks (as words) extended by an edge are disjunct if each set contains extensions of distinct walks. (Contributed by Alexander van der Vekens, 29-Jul-2018.)
Assertion
Ref Expression
disjxwwlks  |- Disj  y  e.  ( ( V WWalksN  E
) `  N ) { x  e. Word  V  | 
( ( x substr  <. 0 ,  N >. )  =  y  /\  ( y ` 
0 )  =  P  /\  { ( lastS  `  y
) ,  ( lastS  `  x
) }  e.  ran  E ) }
Distinct variable groups:    x, E, y    x, N, y    x, V, y
Allowed substitution hints:    P( x, y)

Proof of Theorem disjxwwlks
StepHypRef Expression
1 simp1 996 . . . . 5  |-  ( ( ( x substr  <. 0 ,  N >. )  =  y  /\  ( y ` 
0 )  =  P  /\  { ( lastS  `  y
) ,  ( lastS  `  x
) }  e.  ran  E )  ->  ( x substr  <.
0 ,  N >. )  =  y )
21a1i 11 . . . 4  |-  ( x  e. Word  V  ->  (
( ( x substr  <. 0 ,  N >. )  =  y  /\  ( y ` 
0 )  =  P  /\  { ( lastS  `  y
) ,  ( lastS  `  x
) }  e.  ran  E )  ->  ( x substr  <.
0 ,  N >. )  =  y ) )
32ss2rabi 3582 . . 3  |-  { x  e. Word  V  |  ( ( x substr  <. 0 ,  N >. )  =  y  /\  ( y `  0
)  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E
) }  C_  { x  e. Word  V  |  ( x substr  <. 0 ,  N >. )  =  y }
43rgenw 2825 . 2  |-  A. y  e.  ( ( V WWalksN  E
) `  N ) { x  e. Word  V  | 
( ( x substr  <. 0 ,  N >. )  =  y  /\  ( y ` 
0 )  =  P  /\  { ( lastS  `  y
) ,  ( lastS  `  x
) }  e.  ran  E ) }  C_  { x  e. Word  V  |  ( x substr  <. 0 ,  N >. )  =  y }
5 disjxwrd 12643 . 2  |- Disj  y  e.  ( ( V WWalksN  E
) `  N ) { x  e. Word  V  | 
( x substr  <. 0 ,  N >. )  =  y }
6 disjss2 4420 . 2  |-  ( A. y  e.  ( ( V WWalksN  E ) `  N
) { x  e. Word  V  |  ( (
x substr  <. 0 ,  N >. )  =  y  /\  ( y `  0
)  =  P  /\  { ( lastS  `  y ) ,  ( lastS  `  x ) }  e.  ran  E
) }  C_  { x  e. Word  V  |  ( x substr  <. 0 ,  N >. )  =  y }  ->  (Disj  y  e.  ( ( V WWalksN  E ) `  N
) { x  e. Word  V  |  ( x substr  <.
0 ,  N >. )  =  y }  -> Disj  y  e.  ( ( V WWalksN  E
) `  N ) { x  e. Word  V  | 
( ( x substr  <. 0 ,  N >. )  =  y  /\  ( y ` 
0 )  =  P  /\  { ( lastS  `  y
) ,  ( lastS  `  x
) }  e.  ran  E ) } ) )
74, 5, 6mp2 9 1  |- Disj  y  e.  ( ( V WWalksN  E
) `  N ) { x  e. Word  V  | 
( ( x substr  <. 0 ,  N >. )  =  y  /\  ( y ` 
0 )  =  P  /\  { ( lastS  `  y
) ,  ( lastS  `  x
) }  e.  ran  E ) }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818    C_ wss 3476   {cpr 4029   <.cop 4033  Disj wdisj 4417   ran crn 5000   ` cfv 5588  (class class class)co 6284   0cc0 9492  Word cword 12500   lastS clsw 12501   substr csubstr 12504   WWalksN cwwlkn 24382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rmo 2822  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-disj 4418  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6287
This theorem is referenced by: (None)
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