Step |
Hyp |
Ref |
Expression |
1 |
|
clwlkclwwlkf.c |
|- C = { w e. ( ClWalks ` G ) | 1 <_ ( # ` ( 1st ` w ) ) } |
2 |
|
clwlkclwwlkf.f |
|- F = ( c e. C |-> ( ( 2nd ` c ) prefix ( ( # ` ( 2nd ` c ) ) - 1 ) ) ) |
3 |
1 2
|
clwlkclwwlkf |
|- ( G e. USPGraph -> F : C --> ( ClWWalks ` G ) ) |
4 |
|
fveq2 |
|- ( c = x -> ( 2nd ` c ) = ( 2nd ` x ) ) |
5 |
|
2fveq3 |
|- ( c = x -> ( # ` ( 2nd ` c ) ) = ( # ` ( 2nd ` x ) ) ) |
6 |
5
|
oveq1d |
|- ( c = x -> ( ( # ` ( 2nd ` c ) ) - 1 ) = ( ( # ` ( 2nd ` x ) ) - 1 ) ) |
7 |
4 6
|
oveq12d |
|- ( c = x -> ( ( 2nd ` c ) prefix ( ( # ` ( 2nd ` c ) ) - 1 ) ) = ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) ) |
8 |
|
id |
|- ( x e. C -> x e. C ) |
9 |
|
ovexd |
|- ( x e. C -> ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) e. _V ) |
10 |
2 7 8 9
|
fvmptd3 |
|- ( x e. C -> ( F ` x ) = ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) ) |
11 |
|
fveq2 |
|- ( c = y -> ( 2nd ` c ) = ( 2nd ` y ) ) |
12 |
|
2fveq3 |
|- ( c = y -> ( # ` ( 2nd ` c ) ) = ( # ` ( 2nd ` y ) ) ) |
13 |
12
|
oveq1d |
|- ( c = y -> ( ( # ` ( 2nd ` c ) ) - 1 ) = ( ( # ` ( 2nd ` y ) ) - 1 ) ) |
14 |
11 13
|
oveq12d |
|- ( c = y -> ( ( 2nd ` c ) prefix ( ( # ` ( 2nd ` c ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) ) |
15 |
|
id |
|- ( y e. C -> y e. C ) |
16 |
|
ovexd |
|- ( y e. C -> ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) e. _V ) |
17 |
2 14 15 16
|
fvmptd3 |
|- ( y e. C -> ( F ` y ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) ) |
18 |
10 17
|
eqeqan12d |
|- ( ( x e. C /\ y e. C ) -> ( ( F ` x ) = ( F ` y ) <-> ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) ) ) |
19 |
18
|
adantl |
|- ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) -> ( ( F ` x ) = ( F ` y ) <-> ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) ) ) |
20 |
|
simplrl |
|- ( ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) /\ ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) ) -> x e. C ) |
21 |
|
simplrr |
|- ( ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) /\ ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) ) -> y e. C ) |
22 |
|
eqid |
|- ( 1st ` x ) = ( 1st ` x ) |
23 |
|
eqid |
|- ( 2nd ` x ) = ( 2nd ` x ) |
24 |
1 22 23
|
clwlkclwwlkflem |
|- ( x e. C -> ( ( 1st ` x ) ( Walks ` G ) ( 2nd ` x ) /\ ( ( 2nd ` x ) ` 0 ) = ( ( 2nd ` x ) ` ( # ` ( 1st ` x ) ) ) /\ ( # ` ( 1st ` x ) ) e. NN ) ) |
25 |
|
wlklenvm1 |
|- ( ( 1st ` x ) ( Walks ` G ) ( 2nd ` x ) -> ( # ` ( 1st ` x ) ) = ( ( # ` ( 2nd ` x ) ) - 1 ) ) |
26 |
25
|
eqcomd |
|- ( ( 1st ` x ) ( Walks ` G ) ( 2nd ` x ) -> ( ( # ` ( 2nd ` x ) ) - 1 ) = ( # ` ( 1st ` x ) ) ) |
27 |
26
|
3ad2ant1 |
|- ( ( ( 1st ` x ) ( Walks ` G ) ( 2nd ` x ) /\ ( ( 2nd ` x ) ` 0 ) = ( ( 2nd ` x ) ` ( # ` ( 1st ` x ) ) ) /\ ( # ` ( 1st ` x ) ) e. NN ) -> ( ( # ` ( 2nd ` x ) ) - 1 ) = ( # ` ( 1st ` x ) ) ) |
28 |
24 27
|
syl |
|- ( x e. C -> ( ( # ` ( 2nd ` x ) ) - 1 ) = ( # ` ( 1st ` x ) ) ) |
29 |
28
|
adantr |
|- ( ( x e. C /\ y e. C ) -> ( ( # ` ( 2nd ` x ) ) - 1 ) = ( # ` ( 1st ` x ) ) ) |
30 |
29
|
oveq2d |
|- ( ( x e. C /\ y e. C ) -> ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) = ( ( 2nd ` x ) prefix ( # ` ( 1st ` x ) ) ) ) |
31 |
|
eqid |
|- ( 1st ` y ) = ( 1st ` y ) |
32 |
|
eqid |
|- ( 2nd ` y ) = ( 2nd ` y ) |
33 |
1 31 32
|
clwlkclwwlkflem |
|- ( y e. C -> ( ( 1st ` y ) ( Walks ` G ) ( 2nd ` y ) /\ ( ( 2nd ` y ) ` 0 ) = ( ( 2nd ` y ) ` ( # ` ( 1st ` y ) ) ) /\ ( # ` ( 1st ` y ) ) e. NN ) ) |
34 |
|
wlklenvm1 |
|- ( ( 1st ` y ) ( Walks ` G ) ( 2nd ` y ) -> ( # ` ( 1st ` y ) ) = ( ( # ` ( 2nd ` y ) ) - 1 ) ) |
35 |
34
|
eqcomd |
|- ( ( 1st ` y ) ( Walks ` G ) ( 2nd ` y ) -> ( ( # ` ( 2nd ` y ) ) - 1 ) = ( # ` ( 1st ` y ) ) ) |
36 |
35
|
3ad2ant1 |
|- ( ( ( 1st ` y ) ( Walks ` G ) ( 2nd ` y ) /\ ( ( 2nd ` y ) ` 0 ) = ( ( 2nd ` y ) ` ( # ` ( 1st ` y ) ) ) /\ ( # ` ( 1st ` y ) ) e. NN ) -> ( ( # ` ( 2nd ` y ) ) - 1 ) = ( # ` ( 1st ` y ) ) ) |
37 |
33 36
|
syl |
|- ( y e. C -> ( ( # ` ( 2nd ` y ) ) - 1 ) = ( # ` ( 1st ` y ) ) ) |
38 |
37
|
adantl |
|- ( ( x e. C /\ y e. C ) -> ( ( # ` ( 2nd ` y ) ) - 1 ) = ( # ` ( 1st ` y ) ) ) |
39 |
38
|
oveq2d |
|- ( ( x e. C /\ y e. C ) -> ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( # ` ( 1st ` y ) ) ) ) |
40 |
30 39
|
eqeq12d |
|- ( ( x e. C /\ y e. C ) -> ( ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) <-> ( ( 2nd ` x ) prefix ( # ` ( 1st ` x ) ) ) = ( ( 2nd ` y ) prefix ( # ` ( 1st ` y ) ) ) ) ) |
41 |
40
|
adantl |
|- ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) -> ( ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) <-> ( ( 2nd ` x ) prefix ( # ` ( 1st ` x ) ) ) = ( ( 2nd ` y ) prefix ( # ` ( 1st ` y ) ) ) ) ) |
42 |
41
|
biimpa |
|- ( ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) /\ ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) ) -> ( ( 2nd ` x ) prefix ( # ` ( 1st ` x ) ) ) = ( ( 2nd ` y ) prefix ( # ` ( 1st ` y ) ) ) ) |
43 |
20 21 42
|
3jca |
|- ( ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) /\ ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) ) -> ( x e. C /\ y e. C /\ ( ( 2nd ` x ) prefix ( # ` ( 1st ` x ) ) ) = ( ( 2nd ` y ) prefix ( # ` ( 1st ` y ) ) ) ) ) |
44 |
1 22 23 31 32
|
clwlkclwwlkf1lem2 |
|- ( ( x e. C /\ y e. C /\ ( ( 2nd ` x ) prefix ( # ` ( 1st ` x ) ) ) = ( ( 2nd ` y ) prefix ( # ` ( 1st ` y ) ) ) ) -> ( ( # ` ( 1st ` x ) ) = ( # ` ( 1st ` y ) ) /\ A. i e. ( 0 ..^ ( # ` ( 1st ` x ) ) ) ( ( 2nd ` x ) ` i ) = ( ( 2nd ` y ) ` i ) ) ) |
45 |
|
simpl |
|- ( ( ( # ` ( 1st ` x ) ) = ( # ` ( 1st ` y ) ) /\ A. i e. ( 0 ..^ ( # ` ( 1st ` x ) ) ) ( ( 2nd ` x ) ` i ) = ( ( 2nd ` y ) ` i ) ) -> ( # ` ( 1st ` x ) ) = ( # ` ( 1st ` y ) ) ) |
46 |
43 44 45
|
3syl |
|- ( ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) /\ ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) ) -> ( # ` ( 1st ` x ) ) = ( # ` ( 1st ` y ) ) ) |
47 |
1 22 23 31 32
|
clwlkclwwlkf1lem3 |
|- ( ( x e. C /\ y e. C /\ ( ( 2nd ` x ) prefix ( # ` ( 1st ` x ) ) ) = ( ( 2nd ` y ) prefix ( # ` ( 1st ` y ) ) ) ) -> A. i e. ( 0 ... ( # ` ( 1st ` x ) ) ) ( ( 2nd ` x ) ` i ) = ( ( 2nd ` y ) ` i ) ) |
48 |
43 47
|
syl |
|- ( ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) /\ ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) ) -> A. i e. ( 0 ... ( # ` ( 1st ` x ) ) ) ( ( 2nd ` x ) ` i ) = ( ( 2nd ` y ) ` i ) ) |
49 |
|
simpl |
|- ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) -> G e. USPGraph ) |
50 |
|
wlkcpr |
|- ( x e. ( Walks ` G ) <-> ( 1st ` x ) ( Walks ` G ) ( 2nd ` x ) ) |
51 |
50
|
biimpri |
|- ( ( 1st ` x ) ( Walks ` G ) ( 2nd ` x ) -> x e. ( Walks ` G ) ) |
52 |
51
|
3ad2ant1 |
|- ( ( ( 1st ` x ) ( Walks ` G ) ( 2nd ` x ) /\ ( ( 2nd ` x ) ` 0 ) = ( ( 2nd ` x ) ` ( # ` ( 1st ` x ) ) ) /\ ( # ` ( 1st ` x ) ) e. NN ) -> x e. ( Walks ` G ) ) |
53 |
24 52
|
syl |
|- ( x e. C -> x e. ( Walks ` G ) ) |
54 |
|
wlkcpr |
|- ( y e. ( Walks ` G ) <-> ( 1st ` y ) ( Walks ` G ) ( 2nd ` y ) ) |
55 |
54
|
biimpri |
|- ( ( 1st ` y ) ( Walks ` G ) ( 2nd ` y ) -> y e. ( Walks ` G ) ) |
56 |
55
|
3ad2ant1 |
|- ( ( ( 1st ` y ) ( Walks ` G ) ( 2nd ` y ) /\ ( ( 2nd ` y ) ` 0 ) = ( ( 2nd ` y ) ` ( # ` ( 1st ` y ) ) ) /\ ( # ` ( 1st ` y ) ) e. NN ) -> y e. ( Walks ` G ) ) |
57 |
33 56
|
syl |
|- ( y e. C -> y e. ( Walks ` G ) ) |
58 |
53 57
|
anim12i |
|- ( ( x e. C /\ y e. C ) -> ( x e. ( Walks ` G ) /\ y e. ( Walks ` G ) ) ) |
59 |
58
|
adantl |
|- ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) -> ( x e. ( Walks ` G ) /\ y e. ( Walks ` G ) ) ) |
60 |
|
eqidd |
|- ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) -> ( # ` ( 1st ` x ) ) = ( # ` ( 1st ` x ) ) ) |
61 |
49 59 60
|
3jca |
|- ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) -> ( G e. USPGraph /\ ( x e. ( Walks ` G ) /\ y e. ( Walks ` G ) ) /\ ( # ` ( 1st ` x ) ) = ( # ` ( 1st ` x ) ) ) ) |
62 |
61
|
adantr |
|- ( ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) /\ ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) ) -> ( G e. USPGraph /\ ( x e. ( Walks ` G ) /\ y e. ( Walks ` G ) ) /\ ( # ` ( 1st ` x ) ) = ( # ` ( 1st ` x ) ) ) ) |
63 |
|
uspgr2wlkeq |
|- ( ( G e. USPGraph /\ ( x e. ( Walks ` G ) /\ y e. ( Walks ` G ) ) /\ ( # ` ( 1st ` x ) ) = ( # ` ( 1st ` x ) ) ) -> ( x = y <-> ( ( # ` ( 1st ` x ) ) = ( # ` ( 1st ` y ) ) /\ A. i e. ( 0 ... ( # ` ( 1st ` x ) ) ) ( ( 2nd ` x ) ` i ) = ( ( 2nd ` y ) ` i ) ) ) ) |
64 |
62 63
|
syl |
|- ( ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) /\ ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) ) -> ( x = y <-> ( ( # ` ( 1st ` x ) ) = ( # ` ( 1st ` y ) ) /\ A. i e. ( 0 ... ( # ` ( 1st ` x ) ) ) ( ( 2nd ` x ) ` i ) = ( ( 2nd ` y ) ` i ) ) ) ) |
65 |
46 48 64
|
mpbir2and |
|- ( ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) /\ ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) ) -> x = y ) |
66 |
65
|
ex |
|- ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) -> ( ( ( 2nd ` x ) prefix ( ( # ` ( 2nd ` x ) ) - 1 ) ) = ( ( 2nd ` y ) prefix ( ( # ` ( 2nd ` y ) ) - 1 ) ) -> x = y ) ) |
67 |
19 66
|
sylbid |
|- ( ( G e. USPGraph /\ ( x e. C /\ y e. C ) ) -> ( ( F ` x ) = ( F ` y ) -> x = y ) ) |
68 |
67
|
ralrimivva |
|- ( G e. USPGraph -> A. x e. C A. y e. C ( ( F ` x ) = ( F ` y ) -> x = y ) ) |
69 |
|
dff13 |
|- ( F : C -1-1-> ( ClWWalks ` G ) <-> ( F : C --> ( ClWWalks ` G ) /\ A. x e. C A. y e. C ( ( F ` x ) = ( F ` y ) -> x = y ) ) ) |
70 |
3 68 69
|
sylanbrc |
|- ( G e. USPGraph -> F : C -1-1-> ( ClWWalks ` G ) ) |