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Theorem disjxwwlks 24938
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 994 . . . . 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 3568 . . 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 2815 . 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 12671 . 2  |- Disj  y  e.  ( ( V WWalksN  E
) `  N ) { x  e. Word  V  | 
( x substr  <. 0 ,  N >. )  =  y }
6 disjss2 4413 . 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 971    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808    C_ wss 3461   {cpr 4018   <.cop 4022  Disj wdisj 4410   ran crn 4989   ` cfv 5570  (class class class)co 6270   0cc0 9481  Word cword 12518   lastS clsw 12519   substr csubstr 12522   WWalksN cwwlkn 24880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-in 3468  df-ss 3475  df-disj 4411
This theorem is referenced by: (None)
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