Theorem disjxwwlksn 26799

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.) (Revised by AV, 19-Apr-2021.)

Hypotheses

Ref	Expression
wwlksnexthasheq.v	|- V = (Vtx ` G )
wwlksnexthasheq.e	|- E = (Edg ` G )

Assertion

Ref	Expression
disjxwwlksn	|- Disj y e. ( N WWalksN G ) { x e. Word V | ( ( x substr <. 0 , N >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) }

Distinct variable groups:   y, N   x, V   x, y

Allowed substitution hints:   P( x, y)   E( x, y)   G( x, y)   N( x)   V( y)

Proof of Theorem disjxwwlksn

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . 5 |- ( ( ( x substr <. 0 , N >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) -> ( x substr <. 0 , N >. ) = y )
2	1	a1i 11	. . . 4 |- ( x e. Word V -> ( ( ( x substr <. 0 , N >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) -> ( x substr <. 0 , N >. ) = y ) )
3	2	ss2rabi 3684	. . 3 |- { x e. Word V | ( ( x substr <. 0 , N >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) } C_ { x e. Word V | ( x substr <. 0 , N >. ) = y }
4	3	rgenw 2924	. 2 |- A. y e. ( N WWalksN G ) { x e. Word V | ( ( x substr <. 0 , N >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) } C_ { x e. Word V | ( x substr <. 0 , N >. ) = y }
5		disjxwrd 13455	. 2 |- Disj y e. ( N WWalksN G ) { x e. Word V | ( x substr <. 0 , N >. ) = y }
6		disjss2 4623	. 2 |- ( A. y e. ( N WWalksN G ) { x e. Word V | ( ( x substr <. 0 , N >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) } C_ { x e. Word V | ( x substr <. 0 , N >. ) = y } -> (Disj y e. ( N WWalksN G ) { x e. Word V | ( x substr <. 0 , N >. ) = y } -> Disj y e. ( N WWalksN G ) { x e. Word V | ( ( x substr <. 0 , N >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) } ) )
7	4, 5, 6	mp2 9	1 |- Disj y e. ( N WWalksN G ) { x e. Word V | ( ( x substr <. 0 , N >. ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) }

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   {cpr 4179   <.cop 4183  Disj wdisj 4620   ` cfv 5888  (class class class)co 6650   0cc0 9936  Word cword 13291   lastS clsw 13292   substr csubstr 13295  Vtxcvtx 25874  Edgcedg 25939   WWalksN cwwlksn 26718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-in 3581  df-ss 3588  df-disj 4621

This theorem is referenced by: (None)