Step |
Hyp |
Ref |
Expression |
1 |
|
wlkv |
|- ( F ( Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) ) |
2 |
|
simp3l |
|- ( ( G e. _V /\ F ( Walks ` G ) P /\ ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> Fun ( iEdg ` G ) ) |
3 |
|
simp2 |
|- ( ( G e. _V /\ F ( Walks ` G ) P /\ ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> F ( Walks ` G ) P ) |
4 |
|
c0ex |
|- 0 e. _V |
5 |
4
|
snid |
|- 0 e. { 0 } |
6 |
|
oveq2 |
|- ( ( # ` F ) = 1 -> ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 1 ) ) |
7 |
|
fzo01 |
|- ( 0 ..^ 1 ) = { 0 } |
8 |
6 7
|
eqtrdi |
|- ( ( # ` F ) = 1 -> ( 0 ..^ ( # ` F ) ) = { 0 } ) |
9 |
5 8
|
eleqtrrid |
|- ( ( # ` F ) = 1 -> 0 e. ( 0 ..^ ( # ` F ) ) ) |
10 |
9
|
ad2antrl |
|- ( ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> 0 e. ( 0 ..^ ( # ` F ) ) ) |
11 |
10
|
3ad2ant3 |
|- ( ( G e. _V /\ F ( Walks ` G ) P /\ ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> 0 e. ( 0 ..^ ( # ` F ) ) ) |
12 |
|
eqid |
|- ( iEdg ` G ) = ( iEdg ` G ) |
13 |
12
|
iedginwlk |
|- ( ( Fun ( iEdg ` G ) /\ F ( Walks ` G ) P /\ 0 e. ( 0 ..^ ( # ` F ) ) ) -> ( ( iEdg ` G ) ` ( F ` 0 ) ) e. ran ( iEdg ` G ) ) |
14 |
2 3 11 13
|
syl3anc |
|- ( ( G e. _V /\ F ( Walks ` G ) P /\ ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> ( ( iEdg ` G ) ` ( F ` 0 ) ) e. ran ( iEdg ` G ) ) |
15 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
16 |
15 12
|
iswlkg |
|- ( G 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 ) ) ) ) ) ) |
17 |
8
|
raleqdv |
|- ( ( # ` F ) = 1 -> ( 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 } if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) ) ) |
18 |
|
oveq1 |
|- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) ) |
19 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
20 |
18 19
|
eqtrdi |
|- ( k = 0 -> ( k + 1 ) = 1 ) |
21 |
|
wkslem2 |
|- ( ( k = 0 /\ ( k + 1 ) = 1 ) -> ( if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( ( iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( ( iEdg ` G ) ` ( F ` 0 ) ) ) ) ) |
22 |
20 21
|
mpdan |
|- ( k = 0 -> ( if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( ( iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( ( iEdg ` G ) ` ( F ` 0 ) ) ) ) ) |
23 |
4 22
|
ralsn |
|- ( A. k e. { 0 } if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( F ` k ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( ( iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( ( iEdg ` G ) ` ( F ` 0 ) ) ) ) |
24 |
17 23
|
bitrdi |
|- ( ( # ` F ) = 1 -> ( 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 ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( ( iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( ( iEdg ` G ) ` ( F ` 0 ) ) ) ) ) |
25 |
24
|
ad2antrl |
|- ( ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> ( 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 ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( ( iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( ( iEdg ` G ) ` ( F ` 0 ) ) ) ) ) |
26 |
|
ifptru |
|- ( ( P ` 0 ) = ( P ` 1 ) -> ( if- ( ( P ` 0 ) = ( P ` 1 ) , ( ( iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( ( iEdg ` G ) ` ( F ` 0 ) ) ) <-> ( ( iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } ) ) |
27 |
26
|
biimpa |
|- ( ( ( P ` 0 ) = ( P ` 1 ) /\ if- ( ( P ` 0 ) = ( P ` 1 ) , ( ( iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( ( iEdg ` G ) ` ( F ` 0 ) ) ) ) -> ( ( iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } ) |
28 |
27
|
eqcomd |
|- ( ( ( P ` 0 ) = ( P ` 1 ) /\ if- ( ( P ` 0 ) = ( P ` 1 ) , ( ( iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( ( iEdg ` G ) ` ( F ` 0 ) ) ) ) -> { ( P ` 0 ) } = ( ( iEdg ` G ) ` ( F ` 0 ) ) ) |
29 |
28
|
ex |
|- ( ( P ` 0 ) = ( P ` 1 ) -> ( if- ( ( P ` 0 ) = ( P ` 1 ) , ( ( iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( ( iEdg ` G ) ` ( F ` 0 ) ) ) -> { ( P ` 0 ) } = ( ( iEdg ` G ) ` ( F ` 0 ) ) ) ) |
30 |
29
|
ad2antll |
|- ( ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> ( if- ( ( P ` 0 ) = ( P ` 1 ) , ( ( iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( ( iEdg ` G ) ` ( F ` 0 ) ) ) -> { ( P ` 0 ) } = ( ( iEdg ` G ) ` ( F ` 0 ) ) ) ) |
31 |
25 30
|
sylbid |
|- ( ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> ( 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 ) } = ( ( iEdg ` G ) ` ( F ` 0 ) ) ) ) |
32 |
31
|
com12 |
|- ( 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 ) ) ) -> ( ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } = ( ( iEdg ` G ) ` ( F ` 0 ) ) ) ) |
33 |
32
|
3ad2ant3 |
|- ( ( 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 ) ) ) ) -> ( ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } = ( ( iEdg ` G ) ` ( F ` 0 ) ) ) ) |
34 |
16 33
|
syl6bi |
|- ( G e. _V -> ( F ( Walks ` G ) P -> ( ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } = ( ( iEdg ` G ) ` ( F ` 0 ) ) ) ) ) |
35 |
34
|
3imp |
|- ( ( G e. _V /\ F ( Walks ` G ) P /\ ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> { ( P ` 0 ) } = ( ( iEdg ` G ) ` ( F ` 0 ) ) ) |
36 |
|
edgval |
|- ( Edg ` G ) = ran ( iEdg ` G ) |
37 |
36
|
a1i |
|- ( ( G e. _V /\ F ( Walks ` G ) P /\ ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> ( Edg ` G ) = ran ( iEdg ` G ) ) |
38 |
14 35 37
|
3eltr4d |
|- ( ( G e. _V /\ F ( Walks ` G ) P /\ ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> { ( P ` 0 ) } e. ( Edg ` G ) ) |
39 |
38
|
3exp |
|- ( G e. _V -> ( F ( Walks ` G ) P -> ( ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } e. ( Edg ` G ) ) ) ) |
40 |
39
|
3ad2ant1 |
|- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F ( Walks ` G ) P -> ( ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } e. ( Edg ` G ) ) ) ) |
41 |
1 40
|
mpcom |
|- ( F ( Walks ` G ) P -> ( ( Fun ( iEdg ` G ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } e. ( Edg ` G ) ) ) |
42 |
41
|
expd |
|- ( F ( Walks ` G ) P -> ( Fun ( iEdg ` G ) -> ( ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) -> { ( P ` 0 ) } e. ( Edg ` G ) ) ) ) |
43 |
42
|
impcom |
|- ( ( Fun ( iEdg ` G ) /\ F ( Walks ` G ) P ) -> ( ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) -> { ( P ` 0 ) } e. ( Edg ` G ) ) ) |
44 |
43
|
imp |
|- ( ( ( Fun ( iEdg ` G ) /\ F ( Walks ` G ) P ) /\ ( ( # ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } e. ( Edg ` G ) ) |