Step |
Hyp |
Ref |
Expression |
1 |
|
ispth |
|- ( F ( Paths ` G ) P <-> ( F ( Trails ` G ) P /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) ) |
2 |
|
trliswlk |
|- ( F ( Trails ` G ) P -> F ( Walks ` G ) P ) |
3 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
4 |
3
|
wlkp |
|- ( F ( Walks ` G ) P -> P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) ) |
5 |
|
elfz0lmr |
|- ( J e. ( 0 ... ( # ` F ) ) -> ( J = 0 \/ J e. ( 1 ..^ ( # ` F ) ) \/ J = ( # ` F ) ) ) |
6 |
|
elfzo1 |
|- ( I e. ( 1 ..^ ( # ` F ) ) <-> ( I e. NN /\ ( # ` F ) e. NN /\ I < ( # ` F ) ) ) |
7 |
|
nnnn0 |
|- ( ( # ` F ) e. NN -> ( # ` F ) e. NN0 ) |
8 |
7
|
3ad2ant2 |
|- ( ( I e. NN /\ ( # ` F ) e. NN /\ I < ( # ` F ) ) -> ( # ` F ) e. NN0 ) |
9 |
6 8
|
sylbi |
|- ( I e. ( 1 ..^ ( # ` F ) ) -> ( # ` F ) e. NN0 ) |
10 |
9
|
adantl |
|- ( ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( # ` F ) e. NN0 ) |
11 |
|
fvinim0ffz |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( # ` F ) e. NN0 ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) <-> ( ( P ` 0 ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) /\ ( P ` ( # ` F ) ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) ) ) ) |
12 |
10 11
|
sylan2 |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) <-> ( ( P ` 0 ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) /\ ( P ` ( # ` F ) ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) ) ) ) |
13 |
|
fveq2 |
|- ( J = 0 -> ( P ` J ) = ( P ` 0 ) ) |
14 |
13
|
eqeq2d |
|- ( J = 0 -> ( ( P ` I ) = ( P ` J ) <-> ( P ` I ) = ( P ` 0 ) ) ) |
15 |
14
|
ad2antrl |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( P ` I ) = ( P ` J ) <-> ( P ` I ) = ( P ` 0 ) ) ) |
16 |
|
ffun |
|- ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) -> Fun P ) |
17 |
16
|
adantr |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ I e. ( 1 ..^ ( # ` F ) ) ) -> Fun P ) |
18 |
|
fdm |
|- ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) -> dom P = ( 0 ... ( # ` F ) ) ) |
19 |
|
fzo0ss1 |
|- ( 1 ..^ ( # ` F ) ) C_ ( 0 ..^ ( # ` F ) ) |
20 |
|
fzossfz |
|- ( 0 ..^ ( # ` F ) ) C_ ( 0 ... ( # ` F ) ) |
21 |
19 20
|
sstri |
|- ( 1 ..^ ( # ` F ) ) C_ ( 0 ... ( # ` F ) ) |
22 |
21
|
sseli |
|- ( I e. ( 1 ..^ ( # ` F ) ) -> I e. ( 0 ... ( # ` F ) ) ) |
23 |
|
eleq2 |
|- ( dom P = ( 0 ... ( # ` F ) ) -> ( I e. dom P <-> I e. ( 0 ... ( # ` F ) ) ) ) |
24 |
22 23
|
syl5ibr |
|- ( dom P = ( 0 ... ( # ` F ) ) -> ( I e. ( 1 ..^ ( # ` F ) ) -> I e. dom P ) ) |
25 |
18 24
|
syl |
|- ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) -> ( I e. ( 1 ..^ ( # ` F ) ) -> I e. dom P ) ) |
26 |
25
|
imp |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ I e. ( 1 ..^ ( # ` F ) ) ) -> I e. dom P ) |
27 |
17 26
|
jca |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( Fun P /\ I e. dom P ) ) |
28 |
27
|
adantrl |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( Fun P /\ I e. dom P ) ) |
29 |
|
simprr |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> I e. ( 1 ..^ ( # ` F ) ) ) |
30 |
|
funfvima |
|- ( ( Fun P /\ I e. dom P ) -> ( I e. ( 1 ..^ ( # ` F ) ) -> ( P ` I ) e. ( P " ( 1 ..^ ( # ` F ) ) ) ) ) |
31 |
28 29 30
|
sylc |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( P ` I ) e. ( P " ( 1 ..^ ( # ` F ) ) ) ) |
32 |
|
eleq1 |
|- ( ( P ` I ) = ( P ` 0 ) -> ( ( P ` I ) e. ( P " ( 1 ..^ ( # ` F ) ) ) <-> ( P ` 0 ) e. ( P " ( 1 ..^ ( # ` F ) ) ) ) ) |
33 |
31 32
|
syl5ibcom |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( P ` I ) = ( P ` 0 ) -> ( P ` 0 ) e. ( P " ( 1 ..^ ( # ` F ) ) ) ) ) |
34 |
15 33
|
sylbid |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( P ` I ) = ( P ` J ) -> ( P ` 0 ) e. ( P " ( 1 ..^ ( # ` F ) ) ) ) ) |
35 |
|
nnel |
|- ( -. ( P ` 0 ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) <-> ( P ` 0 ) e. ( P " ( 1 ..^ ( # ` F ) ) ) ) |
36 |
34 35
|
syl6ibr |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( P ` I ) = ( P ` J ) -> -. ( P ` 0 ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) ) ) |
37 |
36
|
necon2ad |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( P ` 0 ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) -> ( P ` I ) =/= ( P ` J ) ) ) |
38 |
37
|
adantrd |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( ( P ` 0 ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) /\ ( P ` ( # ` F ) ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) ) -> ( P ` I ) =/= ( P ` J ) ) ) |
39 |
12 38
|
sylbid |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) -> ( P ` I ) =/= ( P ` J ) ) ) |
40 |
39
|
ex |
|- ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) -> ( ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) -> ( P ` I ) =/= ( P ` J ) ) ) ) |
41 |
40
|
com23 |
|- ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) -> ( ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( P ` I ) =/= ( P ` J ) ) ) ) |
42 |
41
|
a1d |
|- ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) -> ( Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) -> ( ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( P ` I ) =/= ( P ` J ) ) ) ) ) |
43 |
42
|
3imp |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( P ` I ) =/= ( P ` J ) ) ) |
44 |
43
|
com12 |
|- ( ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( P ` I ) =/= ( P ` J ) ) ) |
45 |
44
|
a1d |
|- ( ( J = 0 /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( I =/= J -> ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( P ` I ) =/= ( P ` J ) ) ) ) |
46 |
45
|
ex |
|- ( J = 0 -> ( I e. ( 1 ..^ ( # ` F ) ) -> ( I =/= J -> ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( P ` I ) =/= ( P ` J ) ) ) ) ) |
47 |
|
fvres |
|- ( I e. ( 1 ..^ ( # ` F ) ) -> ( ( P |` ( 1 ..^ ( # ` F ) ) ) ` I ) = ( P ` I ) ) |
48 |
47
|
adantl |
|- ( ( J e. ( 1 ..^ ( # ` F ) ) /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( ( P |` ( 1 ..^ ( # ` F ) ) ) ` I ) = ( P ` I ) ) |
49 |
48
|
adantl |
|- ( ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) /\ ( J e. ( 1 ..^ ( # ` F ) ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( P |` ( 1 ..^ ( # ` F ) ) ) ` I ) = ( P ` I ) ) |
50 |
49
|
eqcomd |
|- ( ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) /\ ( J e. ( 1 ..^ ( # ` F ) ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( P ` I ) = ( ( P |` ( 1 ..^ ( # ` F ) ) ) ` I ) ) |
51 |
|
fvres |
|- ( J e. ( 1 ..^ ( # ` F ) ) -> ( ( P |` ( 1 ..^ ( # ` F ) ) ) ` J ) = ( P ` J ) ) |
52 |
51
|
ad2antrl |
|- ( ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) /\ ( J e. ( 1 ..^ ( # ` F ) ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( P |` ( 1 ..^ ( # ` F ) ) ) ` J ) = ( P ` J ) ) |
53 |
52
|
eqcomd |
|- ( ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) /\ ( J e. ( 1 ..^ ( # ` F ) ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( P ` J ) = ( ( P |` ( 1 ..^ ( # ` F ) ) ) ` J ) ) |
54 |
50 53
|
eqeq12d |
|- ( ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) /\ ( J e. ( 1 ..^ ( # ` F ) ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( P ` I ) = ( P ` J ) <-> ( ( P |` ( 1 ..^ ( # ` F ) ) ) ` I ) = ( ( P |` ( 1 ..^ ( # ` F ) ) ) ` J ) ) ) |
55 |
|
fssres |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( 1 ..^ ( # ` F ) ) C_ ( 0 ... ( # ` F ) ) ) -> ( P |` ( 1 ..^ ( # ` F ) ) ) : ( 1 ..^ ( # ` F ) ) --> ( Vtx ` G ) ) |
56 |
21 55
|
mpan2 |
|- ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) -> ( P |` ( 1 ..^ ( # ` F ) ) ) : ( 1 ..^ ( # ` F ) ) --> ( Vtx ` G ) ) |
57 |
|
df-f1 |
|- ( ( P |` ( 1 ..^ ( # ` F ) ) ) : ( 1 ..^ ( # ` F ) ) -1-1-> ( Vtx ` G ) <-> ( ( P |` ( 1 ..^ ( # ` F ) ) ) : ( 1 ..^ ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) ) ) |
58 |
57
|
biimpri |
|- ( ( ( P |` ( 1 ..^ ( # ` F ) ) ) : ( 1 ..^ ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) ) -> ( P |` ( 1 ..^ ( # ` F ) ) ) : ( 1 ..^ ( # ` F ) ) -1-1-> ( Vtx ` G ) ) |
59 |
56 58
|
sylan |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) ) -> ( P |` ( 1 ..^ ( # ` F ) ) ) : ( 1 ..^ ( # ` F ) ) -1-1-> ( Vtx ` G ) ) |
60 |
59
|
3adant3 |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( P |` ( 1 ..^ ( # ` F ) ) ) : ( 1 ..^ ( # ` F ) ) -1-1-> ( Vtx ` G ) ) |
61 |
|
simpr |
|- ( ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) /\ ( J e. ( 1 ..^ ( # ` F ) ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( J e. ( 1 ..^ ( # ` F ) ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) |
62 |
61
|
ancomd |
|- ( ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) /\ ( J e. ( 1 ..^ ( # ` F ) ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( I e. ( 1 ..^ ( # ` F ) ) /\ J e. ( 1 ..^ ( # ` F ) ) ) ) |
63 |
|
f1veqaeq |
|- ( ( ( P |` ( 1 ..^ ( # ` F ) ) ) : ( 1 ..^ ( # ` F ) ) -1-1-> ( Vtx ` G ) /\ ( I e. ( 1 ..^ ( # ` F ) ) /\ J e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( ( P |` ( 1 ..^ ( # ` F ) ) ) ` I ) = ( ( P |` ( 1 ..^ ( # ` F ) ) ) ` J ) -> I = J ) ) |
64 |
60 62 63
|
syl2an2r |
|- ( ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) /\ ( J e. ( 1 ..^ ( # ` F ) ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( ( P |` ( 1 ..^ ( # ` F ) ) ) ` I ) = ( ( P |` ( 1 ..^ ( # ` F ) ) ) ` J ) -> I = J ) ) |
65 |
54 64
|
sylbid |
|- ( ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) /\ ( J e. ( 1 ..^ ( # ` F ) ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( P ` I ) = ( P ` J ) -> I = J ) ) |
66 |
65
|
ancoms |
|- ( ( ( J e. ( 1 ..^ ( # ` F ) ) /\ I e. ( 1 ..^ ( # ` F ) ) ) /\ ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) ) -> ( ( P ` I ) = ( P ` J ) -> I = J ) ) |
67 |
66
|
necon3d |
|- ( ( ( J e. ( 1 ..^ ( # ` F ) ) /\ I e. ( 1 ..^ ( # ` F ) ) ) /\ ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) ) -> ( I =/= J -> ( P ` I ) =/= ( P ` J ) ) ) |
68 |
67
|
ex |
|- ( ( J e. ( 1 ..^ ( # ` F ) ) /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( I =/= J -> ( P ` I ) =/= ( P ` J ) ) ) ) |
69 |
68
|
com23 |
|- ( ( J e. ( 1 ..^ ( # ` F ) ) /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( I =/= J -> ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( P ` I ) =/= ( P ` J ) ) ) ) |
70 |
69
|
ex |
|- ( J e. ( 1 ..^ ( # ` F ) ) -> ( I e. ( 1 ..^ ( # ` F ) ) -> ( I =/= J -> ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( P ` I ) =/= ( P ` J ) ) ) ) ) |
71 |
9
|
adantl |
|- ( ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( # ` F ) e. NN0 ) |
72 |
71 11
|
sylan2 |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) <-> ( ( P ` 0 ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) /\ ( P ` ( # ` F ) ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) ) ) ) |
73 |
|
fveq2 |
|- ( J = ( # ` F ) -> ( P ` J ) = ( P ` ( # ` F ) ) ) |
74 |
73
|
eqeq2d |
|- ( J = ( # ` F ) -> ( ( P ` I ) = ( P ` J ) <-> ( P ` I ) = ( P ` ( # ` F ) ) ) ) |
75 |
74
|
ad2antrl |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( P ` I ) = ( P ` J ) <-> ( P ` I ) = ( P ` ( # ` F ) ) ) ) |
76 |
27
|
adantrl |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( Fun P /\ I e. dom P ) ) |
77 |
|
simprr |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> I e. ( 1 ..^ ( # ` F ) ) ) |
78 |
76 77 30
|
sylc |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( P ` I ) e. ( P " ( 1 ..^ ( # ` F ) ) ) ) |
79 |
|
eleq1 |
|- ( ( P ` I ) = ( P ` ( # ` F ) ) -> ( ( P ` I ) e. ( P " ( 1 ..^ ( # ` F ) ) ) <-> ( P ` ( # ` F ) ) e. ( P " ( 1 ..^ ( # ` F ) ) ) ) ) |
80 |
78 79
|
syl5ibcom |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( P ` I ) = ( P ` ( # ` F ) ) -> ( P ` ( # ` F ) ) e. ( P " ( 1 ..^ ( # ` F ) ) ) ) ) |
81 |
75 80
|
sylbid |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( P ` I ) = ( P ` J ) -> ( P ` ( # ` F ) ) e. ( P " ( 1 ..^ ( # ` F ) ) ) ) ) |
82 |
|
nnel |
|- ( -. ( P ` ( # ` F ) ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) <-> ( P ` ( # ` F ) ) e. ( P " ( 1 ..^ ( # ` F ) ) ) ) |
83 |
81 82
|
syl6ibr |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( P ` I ) = ( P ` J ) -> -. ( P ` ( # ` F ) ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) ) ) |
84 |
83
|
necon2ad |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( P ` ( # ` F ) ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) -> ( P ` I ) =/= ( P ` J ) ) ) |
85 |
84
|
adantld |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( ( P ` 0 ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) /\ ( P ` ( # ` F ) ) e/ ( P " ( 1 ..^ ( # ` F ) ) ) ) -> ( P ` I ) =/= ( P ` J ) ) ) |
86 |
72 85
|
sylbid |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) -> ( P ` I ) =/= ( P ` J ) ) ) |
87 |
86
|
ex |
|- ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) -> ( ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) -> ( P ` I ) =/= ( P ` J ) ) ) ) |
88 |
87
|
com23 |
|- ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) -> ( ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( P ` I ) =/= ( P ` J ) ) ) ) |
89 |
88
|
a1d |
|- ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) -> ( Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) -> ( ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( P ` I ) =/= ( P ` J ) ) ) ) ) |
90 |
89
|
3imp |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( P ` I ) =/= ( P ` J ) ) ) |
91 |
90
|
com12 |
|- ( ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( P ` I ) =/= ( P ` J ) ) ) |
92 |
91
|
a1d |
|- ( ( J = ( # ` F ) /\ I e. ( 1 ..^ ( # ` F ) ) ) -> ( I =/= J -> ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( P ` I ) =/= ( P ` J ) ) ) ) |
93 |
92
|
ex |
|- ( J = ( # ` F ) -> ( I e. ( 1 ..^ ( # ` F ) ) -> ( I =/= J -> ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( P ` I ) =/= ( P ` J ) ) ) ) ) |
94 |
46 70 93
|
3jaoi |
|- ( ( J = 0 \/ J e. ( 1 ..^ ( # ` F ) ) \/ J = ( # ` F ) ) -> ( I e. ( 1 ..^ ( # ` F ) ) -> ( I =/= J -> ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( P ` I ) =/= ( P ` J ) ) ) ) ) |
95 |
5 94
|
syl |
|- ( J e. ( 0 ... ( # ` F ) ) -> ( I e. ( 1 ..^ ( # ` F ) ) -> ( I =/= J -> ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( P ` I ) =/= ( P ` J ) ) ) ) ) |
96 |
95
|
3imp21 |
|- ( ( I e. ( 1 ..^ ( # ` F ) ) /\ J e. ( 0 ... ( # ` F ) ) /\ I =/= J ) -> ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( P ` I ) =/= ( P ` J ) ) ) |
97 |
96
|
com12 |
|- ( ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( ( I e. ( 1 ..^ ( # ` F ) ) /\ J e. ( 0 ... ( # ` F ) ) /\ I =/= J ) -> ( P ` I ) =/= ( P ` J ) ) ) |
98 |
97
|
3exp |
|- ( P : ( 0 ... ( # ` F ) ) --> ( Vtx ` G ) -> ( Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) -> ( ( I e. ( 1 ..^ ( # ` F ) ) /\ J e. ( 0 ... ( # ` F ) ) /\ I =/= J ) -> ( P ` I ) =/= ( P ` J ) ) ) ) ) |
99 |
2 4 98
|
3syl |
|- ( F ( Trails ` G ) P -> ( Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) -> ( ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) -> ( ( I e. ( 1 ..^ ( # ` F ) ) /\ J e. ( 0 ... ( # ` F ) ) /\ I =/= J ) -> ( P ` I ) =/= ( P ` J ) ) ) ) ) |
100 |
99
|
3imp |
|- ( ( F ( Trails ` G ) P /\ Fun `' ( P |` ( 1 ..^ ( # ` F ) ) ) /\ ( ( P " { 0 , ( # ` F ) } ) i^i ( P " ( 1 ..^ ( # ` F ) ) ) ) = (/) ) -> ( ( I e. ( 1 ..^ ( # ` F ) ) /\ J e. ( 0 ... ( # ` F ) ) /\ I =/= J ) -> ( P ` I ) =/= ( P ` J ) ) ) |
101 |
1 100
|
sylbi |
|- ( F ( Paths ` G ) P -> ( ( I e. ( 1 ..^ ( # ` F ) ) /\ J e. ( 0 ... ( # ` F ) ) /\ I =/= J ) -> ( P ` I ) =/= ( P ` J ) ) ) |
102 |
101
|
imp |
|- ( ( F ( Paths ` G ) P /\ ( I e. ( 1 ..^ ( # ` F ) ) /\ J e. ( 0 ... ( # ` F ) ) /\ I =/= J ) ) -> ( P ` I ) =/= ( P ` J ) ) |