Step |
Hyp |
Ref |
Expression |
1 |
|
wlkv |
|- ( F ( Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) ) |
2 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
3 |
|
eqid |
|- ( iEdg ` G ) = ( iEdg ` G ) |
4 |
2 3
|
iswlk |
|- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F ( Walks ` G ) P <-> ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) ) ) |
5 |
|
simpr1 |
|- ( ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) /\ ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) ) -> F e. Word dom ( iEdg ` G ) ) |
6 |
|
simpr2 |
|- ( ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) /\ ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) ) -> P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) |
7 |
|
df-ifp |
|- ( if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) <-> ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } ) \/ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) ) |
8 |
|
dfsn2 |
|- { ( P ` k ) } = { ( P ` k ) , ( P ` k ) } |
9 |
|
preq2 |
|- ( ( P ` k ) = ( P ` ( k + 1 ) ) -> { ( P ` k ) , ( P ` k ) } = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) |
10 |
8 9
|
syl5eq |
|- ( ( P ` k ) = ( P ` ( k + 1 ) ) -> { ( P ` k ) } = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) |
11 |
10
|
eqeq2d |
|- ( ( P ` k ) = ( P ` ( k + 1 ) ) -> ( ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } <-> ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) |
12 |
11
|
biimpa |
|- ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } ) -> ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) |
13 |
12
|
a1d |
|- ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } ) -> ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) -> ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) |
14 |
|
simpr |
|- ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) -> G e. UPGraph ) |
15 |
|
simpl |
|- ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) -> F e. Word dom ( iEdg ` G ) ) |
16 |
|
eqid |
|- ( Edg ` G ) = ( Edg ` G ) |
17 |
3 16
|
upgredginwlk |
|- ( ( G e. UPGraph /\ F e. Word dom ( iEdg ` G ) ) -> ( k e. ( 0 ..^ ( # ` F ) ) -> ( ( iEdg ` G ) ` ( F ` k ) ) e. ( Edg ` G ) ) ) |
18 |
14 15 17
|
syl2anr |
|- ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) -> ( k e. ( 0 ..^ ( # ` F ) ) -> ( ( iEdg ` G ) ` ( F ` k ) ) e. ( Edg ` G ) ) ) |
19 |
18
|
imp |
|- ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) -> ( ( iEdg ` G ) ` ( F ` k ) ) e. ( Edg ` G ) ) |
20 |
|
simprr |
|- ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) -> G e. UPGraph ) |
21 |
20
|
adantr |
|- ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) -> G e. UPGraph ) |
22 |
21
|
adantr |
|- ( ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) /\ ( ( iEdg ` G ) ` ( F ` k ) ) e. ( Edg ` G ) ) -> G e. UPGraph ) |
23 |
22
|
adantr |
|- ( ( ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) /\ ( ( iEdg ` G ) ` ( F ` k ) ) e. ( Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) -> G e. UPGraph ) |
24 |
|
simplr |
|- ( ( ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) /\ ( ( iEdg ` G ) ` ( F ` k ) ) e. ( Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) -> ( ( iEdg ` G ) ` ( F ` k ) ) e. ( Edg ` G ) ) |
25 |
|
simprr |
|- ( ( ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) /\ ( ( iEdg ` G ) ` ( F ` k ) ) e. ( Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) -> { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) |
26 |
|
df-ne |
|- ( ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> -. ( P ` k ) = ( P ` ( k + 1 ) ) ) |
27 |
|
fvexd |
|- ( ( P ` k ) =/= ( P ` ( k + 1 ) ) -> ( P ` k ) e. _V ) |
28 |
|
fvexd |
|- ( ( P ` k ) =/= ( P ` ( k + 1 ) ) -> ( P ` ( k + 1 ) ) e. _V ) |
29 |
|
id |
|- ( ( P ` k ) =/= ( P ` ( k + 1 ) ) -> ( P ` k ) =/= ( P ` ( k + 1 ) ) ) |
30 |
27 28 29
|
3jca |
|- ( ( P ` k ) =/= ( P ` ( k + 1 ) ) -> ( ( P ` k ) e. _V /\ ( P ` ( k + 1 ) ) e. _V /\ ( P ` k ) =/= ( P ` ( k + 1 ) ) ) ) |
31 |
26 30
|
sylbir |
|- ( -. ( P ` k ) = ( P ` ( k + 1 ) ) -> ( ( P ` k ) e. _V /\ ( P ` ( k + 1 ) ) e. _V /\ ( P ` k ) =/= ( P ` ( k + 1 ) ) ) ) |
32 |
31
|
adantr |
|- ( ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) -> ( ( P ` k ) e. _V /\ ( P ` ( k + 1 ) ) e. _V /\ ( P ` k ) =/= ( P ` ( k + 1 ) ) ) ) |
33 |
32
|
adantl |
|- ( ( ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) /\ ( ( iEdg ` G ) ` ( F ` k ) ) e. ( Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) -> ( ( P ` k ) e. _V /\ ( P ` ( k + 1 ) ) e. _V /\ ( P ` k ) =/= ( P ` ( k + 1 ) ) ) ) |
34 |
2 16
|
upgredgpr |
|- ( ( ( G e. UPGraph /\ ( ( iEdg ` G ) ` ( F ` k ) ) e. ( Edg ` G ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) /\ ( ( P ` k ) e. _V /\ ( P ` ( k + 1 ) ) e. _V /\ ( P ` k ) =/= ( P ` ( k + 1 ) ) ) ) -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = ( ( iEdg ` G ) ` ( F ` k ) ) ) |
35 |
23 24 25 33 34
|
syl31anc |
|- ( ( ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) /\ ( ( iEdg ` G ) ` ( F ` k ) ) e. ( Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = ( ( iEdg ` G ) ` ( F ` k ) ) ) |
36 |
35
|
eqcomd |
|- ( ( ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) /\ ( ( iEdg ` G ) ` ( F ` k ) ) e. ( Edg ` G ) ) /\ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) -> ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) |
37 |
36
|
exp31 |
|- ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) -> ( ( ( iEdg ` G ) ` ( F ` k ) ) e. ( Edg ` G ) -> ( ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) -> ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) |
38 |
19 37
|
mpd |
|- ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) -> ( ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) -> ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) |
39 |
38
|
com12 |
|- ( ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) -> ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) -> ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) |
40 |
13 39
|
jaoi |
|- ( ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } ) \/ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) -> ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) -> ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) |
41 |
40
|
com12 |
|- ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) -> ( ( ( ( P ` k ) = ( P ` ( k + 1 ) ) /\ ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } ) \/ ( -. ( P ` k ) = ( P ` ( k + 1 ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) -> ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) |
42 |
7 41
|
syl5bi |
|- ( ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) /\ k e. ( 0 ..^ ( # ` F ) ) ) -> ( if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) -> ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) |
43 |
42
|
ralimdva |
|- ( ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) /\ ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) ) -> ( A. k e. ( 0 ..^ ( # ` F ) ) if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) -> A. k e. ( 0 ..^ ( # ` F ) ) ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) |
44 |
43
|
ex |
|- ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) -> ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) -> ( A. k e. ( 0 ..^ ( # ` F ) ) if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) -> A. k e. ( 0 ..^ ( # ` F ) ) ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) |
45 |
44
|
com23 |
|- ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) -> ( A. k e. ( 0 ..^ ( # ` F ) ) if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) -> ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) -> A. k e. ( 0 ..^ ( # ` F ) ) ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) |
46 |
45
|
3impia |
|- ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) -> ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) -> A. k e. ( 0 ..^ ( # ` F ) ) ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) |
47 |
46
|
impcom |
|- ( ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) /\ ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) ) -> A. k e. ( 0 ..^ ( # ` F ) ) ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) |
48 |
5 6 47
|
3jca |
|- ( ( ( ( G e. _V /\ F e. _V /\ P e. _V ) /\ G e. UPGraph ) /\ ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) ) -> ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) |
49 |
48
|
exp31 |
|- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( G e. UPGraph -> ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) -> ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) ) |
50 |
49
|
com23 |
|- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) -> ( G e. UPGraph -> ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) ) |
51 |
4 50
|
sylbid |
|- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F ( Walks ` G ) P -> ( G e. UPGraph -> ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) ) |
52 |
51
|
impd |
|- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( ( F ( Walks ` G ) P /\ G e. UPGraph ) -> ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) |
53 |
52
|
impcom |
|- ( ( ( F ( Walks ` G ) P /\ G e. UPGraph ) /\ ( G e. _V /\ F e. _V /\ P e. _V ) ) -> ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) |
54 |
2 3
|
isupwlk |
|- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F ( UPWalks ` G ) P <-> ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) |
55 |
54
|
adantl |
|- ( ( ( F ( Walks ` G ) P /\ G e. UPGraph ) /\ ( G e. _V /\ F e. _V /\ P e. _V ) ) -> ( F ( UPWalks ` G ) P <-> ( F e. Word dom ( iEdg ` G ) /\ P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` F ) ) ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) ) |
56 |
53 55
|
mpbird |
|- ( ( ( F ( Walks ` G ) P /\ G e. UPGraph ) /\ ( G e. _V /\ F e. _V /\ P e. _V ) ) -> F ( UPWalks ` G ) P ) |
57 |
56
|
exp31 |
|- ( F ( Walks ` G ) P -> ( G e. UPGraph -> ( ( G e. _V /\ F e. _V /\ P e. _V ) -> F ( UPWalks ` G ) P ) ) ) |
58 |
1 57
|
mpid |
|- ( F ( Walks ` G ) P -> ( G e. UPGraph -> F ( UPWalks ` G ) P ) ) |
59 |
58
|
impcom |
|- ( ( G e. UPGraph /\ F ( Walks ` G ) P ) -> F ( UPWalks ` G ) P ) |