Step |
Hyp |
Ref |
Expression |
1 |
|
3anan32 |
|- ( ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) <-> ( ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) /\ A. y e. ( 0 ..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) ) |
2 |
1
|
a1i |
|- ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) <-> ( ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) /\ A. y e. ( 0 ..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) ) ) |
3 |
|
wlkeq |
|- ( ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) ) ) |
4 |
3
|
3expa |
|- ( ( ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) ) ) |
5 |
4
|
3adant1 |
|- ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) ) ) |
6 |
|
fzofzp1 |
|- ( x e. ( 0 ..^ N ) -> ( x + 1 ) e. ( 0 ... N ) ) |
7 |
6
|
adantl |
|- ( ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ x e. ( 0 ..^ N ) ) -> ( x + 1 ) e. ( 0 ... N ) ) |
8 |
|
fveq2 |
|- ( y = ( x + 1 ) -> ( ( 2nd ` A ) ` y ) = ( ( 2nd ` A ) ` ( x + 1 ) ) ) |
9 |
|
fveq2 |
|- ( y = ( x + 1 ) -> ( ( 2nd ` B ) ` y ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) |
10 |
8 9
|
eqeq12d |
|- ( y = ( x + 1 ) -> ( ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) <-> ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) ) |
11 |
10
|
adantl |
|- ( ( ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ x e. ( 0 ..^ N ) ) /\ y = ( x + 1 ) ) -> ( ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) <-> ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) ) |
12 |
7 11
|
rspcdv |
|- ( ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ x e. ( 0 ..^ N ) ) -> ( A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) -> ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) ) |
13 |
12
|
impancom |
|- ( ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) -> ( x e. ( 0 ..^ N ) -> ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) ) |
14 |
13
|
ralrimiv |
|- ( ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) -> A. x e. ( 0 ..^ N ) ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) |
15 |
|
fvoveq1 |
|- ( y = x -> ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` A ) ` ( x + 1 ) ) ) |
16 |
|
fvoveq1 |
|- ( y = x -> ( ( 2nd ` B ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) |
17 |
15 16
|
eqeq12d |
|- ( y = x -> ( ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) <-> ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) ) |
18 |
17
|
cbvralvw |
|- ( A. y e. ( 0 ..^ N ) ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) <-> A. x e. ( 0 ..^ N ) ( ( 2nd ` A ) ` ( x + 1 ) ) = ( ( 2nd ` B ) ` ( x + 1 ) ) ) |
19 |
14 18
|
sylibr |
|- ( ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) -> A. y e. ( 0 ..^ N ) ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) ) |
20 |
|
fzossfz |
|- ( 0 ..^ N ) C_ ( 0 ... N ) |
21 |
|
ssralv |
|- ( ( 0 ..^ N ) C_ ( 0 ... N ) -> ( A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) -> A. y e. ( 0 ..^ N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) ) |
22 |
20 21
|
mp1i |
|- ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) -> A. y e. ( 0 ..^ N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) ) |
23 |
|
r19.26 |
|- ( A. y e. ( 0 ..^ N ) ( ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) /\ ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) ) <-> ( A. y e. ( 0 ..^ N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) /\ A. y e. ( 0 ..^ N ) ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) ) ) |
24 |
|
preq12 |
|- ( ( ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) /\ ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) ) -> { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) |
25 |
24
|
a1i |
|- ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( ( ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) /\ ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) ) -> { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) |
26 |
25
|
ralimdv |
|- ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ..^ N ) ( ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) /\ ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) ) -> A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) |
27 |
23 26
|
syl5bir |
|- ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( ( A. y e. ( 0 ..^ N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) /\ A. y e. ( 0 ..^ N ) ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) ) -> A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) |
28 |
27
|
expd |
|- ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ..^ N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) -> ( A. y e. ( 0 ..^ N ) ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) -> A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) ) |
29 |
22 28
|
syld |
|- ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) -> ( A. y e. ( 0 ..^ N ) ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) -> A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) ) |
30 |
29
|
imp |
|- ( ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) -> ( A. y e. ( 0 ..^ N ) ( ( 2nd ` A ) ` ( y + 1 ) ) = ( ( 2nd ` B ) ` ( y + 1 ) ) -> A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) |
31 |
19 30
|
mpd |
|- ( ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) -> A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) |
32 |
31
|
ex |
|- ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) -> A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) |
33 |
|
uspgrupgr |
|- ( G e. USPGraph -> G e. UPGraph ) |
34 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
35 |
|
eqid |
|- ( iEdg ` G ) = ( iEdg ` G ) |
36 |
|
eqid |
|- ( 1st ` A ) = ( 1st ` A ) |
37 |
|
eqid |
|- ( 2nd ` A ) = ( 2nd ` A ) |
38 |
34 35 36 37
|
upgrwlkcompim |
|- ( ( G e. UPGraph /\ A e. ( Walks ` G ) ) -> ( ( 1st ` A ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) ) |
39 |
38
|
ex |
|- ( G e. UPGraph -> ( A e. ( Walks ` G ) -> ( ( 1st ` A ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) ) ) |
40 |
33 39
|
syl |
|- ( G e. USPGraph -> ( A e. ( Walks ` G ) -> ( ( 1st ` A ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) ) ) |
41 |
|
eqid |
|- ( 1st ` B ) = ( 1st ` B ) |
42 |
|
eqid |
|- ( 2nd ` B ) = ( 2nd ` B ) |
43 |
34 35 41 42
|
upgrwlkcompim |
|- ( ( G e. UPGraph /\ B e. ( Walks ` G ) ) -> ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) |
44 |
43
|
ex |
|- ( G e. UPGraph -> ( B e. ( Walks ` G ) -> ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) ) |
45 |
33 44
|
syl |
|- ( G e. USPGraph -> ( B e. ( Walks ` G ) -> ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) ) |
46 |
|
oveq2 |
|- ( ( # ` ( 1st ` B ) ) = N -> ( 0 ..^ ( # ` ( 1st ` B ) ) ) = ( 0 ..^ N ) ) |
47 |
46
|
eqcoms |
|- ( N = ( # ` ( 1st ` B ) ) -> ( 0 ..^ ( # ` ( 1st ` B ) ) ) = ( 0 ..^ N ) ) |
48 |
47
|
raleqdv |
|- ( N = ( # ` ( 1st ` B ) ) -> ( A. y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } <-> A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) |
49 |
|
oveq2 |
|- ( ( # ` ( 1st ` A ) ) = N -> ( 0 ..^ ( # ` ( 1st ` A ) ) ) = ( 0 ..^ N ) ) |
50 |
49
|
eqcoms |
|- ( N = ( # ` ( 1st ` A ) ) -> ( 0 ..^ ( # ` ( 1st ` A ) ) ) = ( 0 ..^ N ) ) |
51 |
50
|
raleqdv |
|- ( N = ( # ` ( 1st ` A ) ) -> ( A. y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } <-> A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) ) |
52 |
48 51
|
bi2anan9r |
|- ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( ( A. y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ A. y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) <-> ( A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) ) ) |
53 |
|
r19.26 |
|- ( A. y e. ( 0 ..^ N ) ( ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) <-> ( A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) ) |
54 |
|
eqeq2 |
|- ( { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } <-> ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) |
55 |
|
eqeq2 |
|- ( { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) -> ( ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } <-> ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) |
56 |
55
|
eqcoms |
|- ( ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } <-> ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) |
57 |
56
|
biimpd |
|- ( ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) |
58 |
54 57
|
syl6bi |
|- ( { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } -> ( ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) |
59 |
58
|
com13 |
|- ( ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } -> ( { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) |
60 |
59
|
imp |
|- ( ( ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) -> ( { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) |
61 |
60
|
ral2imi |
|- ( A. y e. ( 0 ..^ N ) ( ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) -> ( A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) |
62 |
53 61
|
sylbir |
|- ( ( A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) -> ( A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) |
63 |
52 62
|
syl6bi |
|- ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( ( A. y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ A. y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) -> ( A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) |
64 |
63
|
com12 |
|- ( ( A. y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } /\ A. y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) -> ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) |
65 |
64
|
ex |
|- ( A. y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> ( A. y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } -> ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) ) |
66 |
65
|
3ad2ant3 |
|- ( ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) -> ( A. y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } -> ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) ) |
67 |
66
|
com12 |
|- ( A. y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } -> ( ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) -> ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) ) |
68 |
67
|
3ad2ant3 |
|- ( ( ( 1st ` A ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) -> ( ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) -> ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) ) |
69 |
68
|
imp |
|- ( ( ( ( 1st ` A ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) /\ ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) -> ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) |
70 |
69
|
expd |
|- ( ( ( ( 1st ` A ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) /\ ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) -> ( N = ( # ` ( 1st ` A ) ) -> ( N = ( # ` ( 1st ` B ) ) -> ( A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) ) |
71 |
70
|
a1i |
|- ( G e. USPGraph -> ( ( ( ( 1st ` A ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } ) /\ ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> ( Vtx ` G ) /\ A. y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } ) ) -> ( N = ( # ` ( 1st ` A ) ) -> ( N = ( # ` ( 1st ` B ) ) -> ( A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) ) ) |
72 |
40 45 71
|
syl2and |
|- ( G e. USPGraph -> ( ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) -> ( N = ( # ` ( 1st ` A ) ) -> ( N = ( # ` ( 1st ` B ) ) -> ( A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) ) ) ) |
73 |
72
|
3imp1 |
|- ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ..^ N ) { ( ( 2nd ` A ) ` y ) , ( ( 2nd ` A ) ` ( y + 1 ) ) } = { ( ( 2nd ` B ) ` y ) , ( ( 2nd ` B ) ` ( y + 1 ) ) } -> A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) ) ) |
74 |
|
eqcom |
|- ( ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) <-> ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) ) |
75 |
35
|
uspgrf1oedg |
|- ( G e. USPGraph -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-onto-> ( Edg ` G ) ) |
76 |
|
f1of1 |
|- ( ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-onto-> ( Edg ` G ) -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> ( Edg ` G ) ) |
77 |
75 76
|
syl |
|- ( G e. USPGraph -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> ( Edg ` G ) ) |
78 |
|
eqidd |
|- ( G e. USPGraph -> ( iEdg ` G ) = ( iEdg ` G ) ) |
79 |
|
eqidd |
|- ( G e. USPGraph -> dom ( iEdg ` G ) = dom ( iEdg ` G ) ) |
80 |
|
edgval |
|- ( Edg ` G ) = ran ( iEdg ` G ) |
81 |
80
|
eqcomi |
|- ran ( iEdg ` G ) = ( Edg ` G ) |
82 |
81
|
a1i |
|- ( G e. USPGraph -> ran ( iEdg ` G ) = ( Edg ` G ) ) |
83 |
78 79 82
|
f1eq123d |
|- ( G e. USPGraph -> ( ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> ran ( iEdg ` G ) <-> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> ( Edg ` G ) ) ) |
84 |
77 83
|
mpbird |
|- ( G e. USPGraph -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> ran ( iEdg ` G ) ) |
85 |
84
|
3ad2ant1 |
|- ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> ran ( iEdg ` G ) ) |
86 |
85
|
adantr |
|- ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> ran ( iEdg ` G ) ) |
87 |
34 35 36 37
|
wlkelwrd |
|- ( A e. ( Walks ` G ) -> ( ( 1st ` A ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> ( Vtx ` G ) ) ) |
88 |
34 35 41 42
|
wlkelwrd |
|- ( B e. ( Walks ` G ) -> ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> ( Vtx ` G ) ) ) |
89 |
|
oveq2 |
|- ( N = ( # ` ( 1st ` A ) ) -> ( 0 ..^ N ) = ( 0 ..^ ( # ` ( 1st ` A ) ) ) ) |
90 |
89
|
eleq2d |
|- ( N = ( # ` ( 1st ` A ) ) -> ( y e. ( 0 ..^ N ) <-> y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ) ) |
91 |
|
wrdsymbcl |
|- ( ( ( 1st ` A ) e. Word dom ( iEdg ` G ) /\ y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) ) -> ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) ) |
92 |
91
|
expcom |
|- ( y e. ( 0 ..^ ( # ` ( 1st ` A ) ) ) -> ( ( 1st ` A ) e. Word dom ( iEdg ` G ) -> ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) ) ) |
93 |
90 92
|
syl6bi |
|- ( N = ( # ` ( 1st ` A ) ) -> ( y e. ( 0 ..^ N ) -> ( ( 1st ` A ) e. Word dom ( iEdg ` G ) -> ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) ) ) ) |
94 |
93
|
adantr |
|- ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( y e. ( 0 ..^ N ) -> ( ( 1st ` A ) e. Word dom ( iEdg ` G ) -> ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) ) ) ) |
95 |
94
|
imp |
|- ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( 1st ` A ) e. Word dom ( iEdg ` G ) -> ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) ) ) |
96 |
95
|
com12 |
|- ( ( 1st ` A ) e. Word dom ( iEdg ` G ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) ) ) |
97 |
96
|
adantl |
|- ( ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 1st ` A ) e. Word dom ( iEdg ` G ) ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) ) ) |
98 |
|
oveq2 |
|- ( N = ( # ` ( 1st ` B ) ) -> ( 0 ..^ N ) = ( 0 ..^ ( # ` ( 1st ` B ) ) ) ) |
99 |
98
|
eleq2d |
|- ( N = ( # ` ( 1st ` B ) ) -> ( y e. ( 0 ..^ N ) <-> y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ) ) |
100 |
|
wrdsymbcl |
|- ( ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) ) -> ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) |
101 |
100
|
expcom |
|- ( y e. ( 0 ..^ ( # ` ( 1st ` B ) ) ) -> ( ( 1st ` B ) e. Word dom ( iEdg ` G ) -> ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) |
102 |
99 101
|
syl6bi |
|- ( N = ( # ` ( 1st ` B ) ) -> ( y e. ( 0 ..^ N ) -> ( ( 1st ` B ) e. Word dom ( iEdg ` G ) -> ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) ) |
103 |
102
|
adantl |
|- ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( y e. ( 0 ..^ N ) -> ( ( 1st ` B ) e. Word dom ( iEdg ` G ) -> ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) ) |
104 |
103
|
imp |
|- ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( 1st ` B ) e. Word dom ( iEdg ` G ) -> ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) |
105 |
104
|
com12 |
|- ( ( 1st ` B ) e. Word dom ( iEdg ` G ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) |
106 |
105
|
adantr |
|- ( ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 1st ` A ) e. Word dom ( iEdg ` G ) ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) |
107 |
97 106
|
jcad |
|- ( ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 1st ` A ) e. Word dom ( iEdg ` G ) ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) ) |
108 |
107
|
ex |
|- ( ( 1st ` B ) e. Word dom ( iEdg ` G ) -> ( ( 1st ` A ) e. Word dom ( iEdg ` G ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) ) ) |
109 |
108
|
adantr |
|- ( ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> ( Vtx ` G ) ) -> ( ( 1st ` A ) e. Word dom ( iEdg ` G ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) ) ) |
110 |
109
|
com12 |
|- ( ( 1st ` A ) e. Word dom ( iEdg ` G ) -> ( ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> ( Vtx ` G ) ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) ) ) |
111 |
110
|
adantr |
|- ( ( ( 1st ` A ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> ( Vtx ` G ) ) -> ( ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> ( Vtx ` G ) ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) ) ) |
112 |
111
|
imp |
|- ( ( ( ( 1st ` A ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` A ) : ( 0 ... ( # ` ( 1st ` A ) ) ) --> ( Vtx ` G ) ) /\ ( ( 1st ` B ) e. Word dom ( iEdg ` G ) /\ ( 2nd ` B ) : ( 0 ... ( # ` ( 1st ` B ) ) ) --> ( Vtx ` G ) ) ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) ) |
113 |
87 88 112
|
syl2an |
|- ( ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) -> ( ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) ) |
114 |
113
|
expd |
|- ( ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) -> ( ( N = ( # ` ( 1st ` A ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( y e. ( 0 ..^ N ) -> ( ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) ) ) |
115 |
114
|
expd |
|- ( ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) -> ( N = ( # ` ( 1st ` A ) ) -> ( N = ( # ` ( 1st ` B ) ) -> ( y e. ( 0 ..^ N ) -> ( ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) ) ) ) |
116 |
115
|
imp |
|- ( ( ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( N = ( # ` ( 1st ` B ) ) -> ( y e. ( 0 ..^ N ) -> ( ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) ) ) |
117 |
116
|
3adant1 |
|- ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( N = ( # ` ( 1st ` B ) ) -> ( y e. ( 0 ..^ N ) -> ( ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) ) ) |
118 |
117
|
imp |
|- ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( y e. ( 0 ..^ N ) -> ( ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) ) |
119 |
118
|
imp |
|- ( ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) |
120 |
|
f1veqaeq |
|- ( ( ( iEdg ` G ) : dom ( iEdg ` G ) -1-1-> ran ( iEdg ` G ) /\ ( ( ( 1st ` A ) ` y ) e. dom ( iEdg ` G ) /\ ( ( 1st ` B ) ` y ) e. dom ( iEdg ` G ) ) ) -> ( ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) -> ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) ) |
121 |
86 119 120
|
syl2an2r |
|- ( ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) -> ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) ) |
122 |
74 121
|
syl5bi |
|- ( ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) /\ y e. ( 0 ..^ N ) ) -> ( ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) -> ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) ) |
123 |
122
|
ralimdva |
|- ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ..^ N ) ( ( iEdg ` G ) ` ( ( 1st ` B ) ` y ) ) = ( ( iEdg ` G ) ` ( ( 1st ` A ) ` y ) ) -> A. y e. ( 0 ..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) ) |
124 |
32 73 123
|
3syld |
|- ( ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) /\ N = ( # ` ( 1st ` B ) ) ) -> ( A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) -> A. y e. ( 0 ..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) ) |
125 |
124
|
expimpd |
|- ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) -> A. y e. ( 0 ..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) ) |
126 |
125
|
pm4.71d |
|- ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) <-> ( ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) /\ A. y e. ( 0 ..^ N ) ( ( 1st ` A ) ` y ) = ( ( 1st ` B ) ` y ) ) ) ) |
127 |
2 5 126
|
3bitr4d |
|- ( ( G e. USPGraph /\ ( A e. ( Walks ` G ) /\ B e. ( Walks ` G ) ) /\ N = ( # ` ( 1st ` A ) ) ) -> ( A = B <-> ( N = ( # ` ( 1st ` B ) ) /\ A. y e. ( 0 ... N ) ( ( 2nd ` A ) ` y ) = ( ( 2nd ` B ) ` y ) ) ) ) |