Step |
Hyp |
Ref |
Expression |
1 |
|
wwlksnextprop.x |
|- X = ( ( N + 1 ) WWalksN G ) |
2 |
|
wwlksnextprop.e |
|- E = ( Edg ` G ) |
3 |
|
wwlksnextprop.y |
|- Y = { w e. ( N WWalksN G ) | ( w ` 0 ) = P } |
4 |
|
simp1 |
|- ( ( ( x prefix M ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) -> ( x prefix M ) = y ) |
5 |
4
|
rgenw |
|- A. x e. X ( ( ( x prefix M ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) -> ( x prefix M ) = y ) |
6 |
|
ss2rab |
|- ( { x e. X | ( ( x prefix M ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) } C_ { x e. X | ( x prefix M ) = y } <-> A. x e. X ( ( ( x prefix M ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) -> ( x prefix M ) = y ) ) |
7 |
5 6
|
mpbir |
|- { x e. X | ( ( x prefix M ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) } C_ { x e. X | ( x prefix M ) = y } |
8 |
|
wwlkssswwlksn |
|- ( ( N + 1 ) WWalksN G ) C_ ( WWalks ` G ) |
9 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
10 |
9
|
wwlkssswrd |
|- ( WWalks ` G ) C_ Word ( Vtx ` G ) |
11 |
8 10
|
sstri |
|- ( ( N + 1 ) WWalksN G ) C_ Word ( Vtx ` G ) |
12 |
1 11
|
eqsstri |
|- X C_ Word ( Vtx ` G ) |
13 |
|
rabss2 |
|- ( X C_ Word ( Vtx ` G ) -> { x e. X | ( x prefix M ) = y } C_ { x e. Word ( Vtx ` G ) | ( x prefix M ) = y } ) |
14 |
12 13
|
ax-mp |
|- { x e. X | ( x prefix M ) = y } C_ { x e. Word ( Vtx ` G ) | ( x prefix M ) = y } |
15 |
7 14
|
sstri |
|- { x e. X | ( ( x prefix M ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) } C_ { x e. Word ( Vtx ` G ) | ( x prefix M ) = y } |
16 |
15
|
rgenw |
|- A. y e. Y { x e. X | ( ( x prefix M ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) } C_ { x e. Word ( Vtx ` G ) | ( x prefix M ) = y } |
17 |
|
disjwrdpfx |
|- Disj_ y e. Y { x e. Word ( Vtx ` G ) | ( x prefix M ) = y } |
18 |
|
disjss2 |
|- ( A. y e. Y { x e. X | ( ( x prefix M ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) } C_ { x e. Word ( Vtx ` G ) | ( x prefix M ) = y } -> ( Disj_ y e. Y { x e. Word ( Vtx ` G ) | ( x prefix M ) = y } -> Disj_ y e. Y { x e. X | ( ( x prefix M ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) } ) ) |
19 |
16 17 18
|
mp2 |
|- Disj_ y e. Y { x e. X | ( ( x prefix M ) = y /\ ( y ` 0 ) = P /\ { ( lastS ` y ) , ( lastS ` x ) } e. E ) } |