Step |
Hyp |
Ref |
Expression |
1 |
|
usgr2pthlem.v |
|- V = ( Vtx ` G ) |
2 |
|
usgr2pthlem.i |
|- I = ( iEdg ` G ) |
3 |
|
usgr2pthspth |
|- ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( F ( Paths ` G ) P <-> F ( SPaths ` G ) P ) ) |
4 |
|
usgrupgr |
|- ( G e. USGraph -> G e. UPGraph ) |
5 |
4
|
adantr |
|- ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> G e. UPGraph ) |
6 |
|
isspth |
|- ( F ( SPaths ` G ) P <-> ( F ( Trails ` G ) P /\ Fun `' P ) ) |
7 |
6
|
a1i |
|- ( G e. UPGraph -> ( F ( SPaths ` G ) P <-> ( F ( Trails ` G ) P /\ Fun `' P ) ) ) |
8 |
1 2
|
upgrf1istrl |
|- ( G e. UPGraph -> ( F ( Trails ` G ) P <-> ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) ) |
9 |
8
|
anbi1d |
|- ( G e. UPGraph -> ( ( F ( Trails ` G ) P /\ Fun `' P ) <-> ( ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ Fun `' P ) ) ) |
10 |
|
oveq2 |
|- ( ( # ` F ) = 2 -> ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 2 ) ) |
11 |
|
f1eq2 |
|- ( ( 0 ..^ ( # ` F ) ) = ( 0 ..^ 2 ) -> ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I <-> F : ( 0 ..^ 2 ) -1-1-> dom I ) ) |
12 |
10 11
|
syl |
|- ( ( # ` F ) = 2 -> ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I <-> F : ( 0 ..^ 2 ) -1-1-> dom I ) ) |
13 |
12
|
biimpd |
|- ( ( # ` F ) = 2 -> ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I -> F : ( 0 ..^ 2 ) -1-1-> dom I ) ) |
14 |
13
|
adantl |
|- ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I -> F : ( 0 ..^ 2 ) -1-1-> dom I ) ) |
15 |
14
|
com12 |
|- ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> F : ( 0 ..^ 2 ) -1-1-> dom I ) ) |
16 |
15
|
3ad2ant1 |
|- ( ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> F : ( 0 ..^ 2 ) -1-1-> dom I ) ) |
17 |
16
|
ad2antrl |
|- ( ( G e. UPGraph /\ ( ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ Fun `' P ) ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> F : ( 0 ..^ 2 ) -1-1-> dom I ) ) |
18 |
|
oveq2 |
|- ( ( # ` F ) = 2 -> ( 0 ... ( # ` F ) ) = ( 0 ... 2 ) ) |
19 |
18
|
feq2d |
|- ( ( # ` F ) = 2 -> ( P : ( 0 ... ( # ` F ) ) --> V <-> P : ( 0 ... 2 ) --> V ) ) |
20 |
|
df-f1 |
|- ( P : ( 0 ... 2 ) -1-1-> V <-> ( P : ( 0 ... 2 ) --> V /\ Fun `' P ) ) |
21 |
20
|
simplbi2 |
|- ( P : ( 0 ... 2 ) --> V -> ( Fun `' P -> P : ( 0 ... 2 ) -1-1-> V ) ) |
22 |
21
|
a1i |
|- ( ( # ` F ) = 2 -> ( P : ( 0 ... 2 ) --> V -> ( Fun `' P -> P : ( 0 ... 2 ) -1-1-> V ) ) ) |
23 |
19 22
|
sylbid |
|- ( ( # ` F ) = 2 -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( Fun `' P -> P : ( 0 ... 2 ) -1-1-> V ) ) ) |
24 |
23
|
adantl |
|- ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( Fun `' P -> P : ( 0 ... 2 ) -1-1-> V ) ) ) |
25 |
24
|
com3l |
|- ( P : ( 0 ... ( # ` F ) ) --> V -> ( Fun `' P -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> P : ( 0 ... 2 ) -1-1-> V ) ) ) |
26 |
25
|
3ad2ant2 |
|- ( ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( Fun `' P -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> P : ( 0 ... 2 ) -1-1-> V ) ) ) |
27 |
26
|
imp |
|- ( ( ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ Fun `' P ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> P : ( 0 ... 2 ) -1-1-> V ) ) |
28 |
27
|
adantl |
|- ( ( G e. UPGraph /\ ( ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ Fun `' P ) ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> P : ( 0 ... 2 ) -1-1-> V ) ) |
29 |
1 2
|
usgr2pthlem |
|- ( ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) ) |
30 |
29
|
ad2antrl |
|- ( ( G e. UPGraph /\ ( ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ Fun `' P ) ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) ) |
31 |
17 28 30
|
3jcad |
|- ( ( G e. UPGraph /\ ( ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ Fun `' P ) ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( F : ( 0 ..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) ) ) |
32 |
31
|
ex |
|- ( G e. UPGraph -> ( ( ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ Fun `' P ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( F : ( 0 ..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) ) ) ) |
33 |
9 32
|
sylbid |
|- ( G e. UPGraph -> ( ( F ( Trails ` G ) P /\ Fun `' P ) -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( F : ( 0 ..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) ) ) ) |
34 |
7 33
|
sylbid |
|- ( G e. UPGraph -> ( F ( SPaths ` G ) P -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( F : ( 0 ..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) ) ) ) |
35 |
34
|
com23 |
|- ( G e. UPGraph -> ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( F ( SPaths ` G ) P -> ( F : ( 0 ..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) ) ) ) |
36 |
5 35
|
mpcom |
|- ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( F ( SPaths ` G ) P -> ( F : ( 0 ..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) ) ) |
37 |
3 36
|
sylbid |
|- ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( F ( Paths ` G ) P -> ( F : ( 0 ..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) ) ) |
38 |
37
|
ex |
|- ( G e. USGraph -> ( ( # ` F ) = 2 -> ( F ( Paths ` G ) P -> ( F : ( 0 ..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) ) ) ) |
39 |
38
|
impcomd |
|- ( G e. USGraph -> ( ( F ( Paths ` G ) P /\ ( # ` F ) = 2 ) -> ( F : ( 0 ..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) ) ) |
40 |
|
2nn0 |
|- 2 e. NN0 |
41 |
|
f1f |
|- ( F : ( 0 ..^ 2 ) -1-1-> dom I -> F : ( 0 ..^ 2 ) --> dom I ) |
42 |
|
fnfzo0hash |
|- ( ( 2 e. NN0 /\ F : ( 0 ..^ 2 ) --> dom I ) -> ( # ` F ) = 2 ) |
43 |
40 41 42
|
sylancr |
|- ( F : ( 0 ..^ 2 ) -1-1-> dom I -> ( # ` F ) = 2 ) |
44 |
|
oveq2 |
|- ( 2 = ( # ` F ) -> ( 0 ..^ 2 ) = ( 0 ..^ ( # ` F ) ) ) |
45 |
44
|
eqcoms |
|- ( ( # ` F ) = 2 -> ( 0 ..^ 2 ) = ( 0 ..^ ( # ` F ) ) ) |
46 |
|
f1eq2 |
|- ( ( 0 ..^ 2 ) = ( 0 ..^ ( # ` F ) ) -> ( F : ( 0 ..^ 2 ) -1-1-> dom I <-> F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I ) ) |
47 |
45 46
|
syl |
|- ( ( # ` F ) = 2 -> ( F : ( 0 ..^ 2 ) -1-1-> dom I <-> F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I ) ) |
48 |
47
|
biimpd |
|- ( ( # ` F ) = 2 -> ( F : ( 0 ..^ 2 ) -1-1-> dom I -> F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I ) ) |
49 |
48
|
imp |
|- ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) -> F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I ) |
50 |
49
|
adantr |
|- ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) -> F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I ) |
51 |
50
|
ad2antrr |
|- ( ( ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) /\ G e. USGraph ) -> F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I ) |
52 |
|
f1f |
|- ( P : ( 0 ... 2 ) -1-1-> V -> P : ( 0 ... 2 ) --> V ) |
53 |
|
oveq2 |
|- ( 2 = ( # ` F ) -> ( 0 ... 2 ) = ( 0 ... ( # ` F ) ) ) |
54 |
53
|
eqcoms |
|- ( ( # ` F ) = 2 -> ( 0 ... 2 ) = ( 0 ... ( # ` F ) ) ) |
55 |
54
|
adantr |
|- ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) -> ( 0 ... 2 ) = ( 0 ... ( # ` F ) ) ) |
56 |
55
|
feq2d |
|- ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) -> ( P : ( 0 ... 2 ) --> V <-> P : ( 0 ... ( # ` F ) ) --> V ) ) |
57 |
52 56
|
syl5ib |
|- ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) -> ( P : ( 0 ... 2 ) -1-1-> V -> P : ( 0 ... ( # ` F ) ) --> V ) ) |
58 |
57
|
imp |
|- ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) -> P : ( 0 ... ( # ` F ) ) --> V ) |
59 |
58
|
ad2antrr |
|- ( ( ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) /\ G e. USGraph ) -> P : ( 0 ... ( # ` F ) ) --> V ) |
60 |
|
eqcom |
|- ( ( P ` 0 ) = x <-> x = ( P ` 0 ) ) |
61 |
60
|
biimpi |
|- ( ( P ` 0 ) = x -> x = ( P ` 0 ) ) |
62 |
61
|
3ad2ant1 |
|- ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) -> x = ( P ` 0 ) ) |
63 |
|
eqcom |
|- ( ( P ` 1 ) = y <-> y = ( P ` 1 ) ) |
64 |
63
|
biimpi |
|- ( ( P ` 1 ) = y -> y = ( P ` 1 ) ) |
65 |
64
|
3ad2ant2 |
|- ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) -> y = ( P ` 1 ) ) |
66 |
62 65
|
preq12d |
|- ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) -> { x , y } = { ( P ` 0 ) , ( P ` 1 ) } ) |
67 |
66
|
eqeq2d |
|- ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) -> ( ( I ` ( F ` 0 ) ) = { x , y } <-> ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) ) |
68 |
67
|
biimpcd |
|- ( ( I ` ( F ` 0 ) ) = { x , y } -> ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) -> ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) ) |
69 |
68
|
adantr |
|- ( ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) -> ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) -> ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) ) |
70 |
69
|
impcom |
|- ( ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) -> ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) |
71 |
|
eqcom |
|- ( ( P ` 2 ) = z <-> z = ( P ` 2 ) ) |
72 |
71
|
biimpi |
|- ( ( P ` 2 ) = z -> z = ( P ` 2 ) ) |
73 |
72
|
3ad2ant3 |
|- ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) -> z = ( P ` 2 ) ) |
74 |
65 73
|
preq12d |
|- ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) -> { y , z } = { ( P ` 1 ) , ( P ` 2 ) } ) |
75 |
74
|
eqeq2d |
|- ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) -> ( ( I ` ( F ` 1 ) ) = { y , z } <-> ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) |
76 |
75
|
biimpcd |
|- ( ( I ` ( F ` 1 ) ) = { y , z } -> ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) -> ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) |
77 |
76
|
adantl |
|- ( ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) -> ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) -> ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) |
78 |
77
|
impcom |
|- ( ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) -> ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) |
79 |
70 78
|
jca |
|- ( ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) -> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) |
80 |
79
|
rexlimivw |
|- ( E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) -> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) |
81 |
80
|
rexlimivw |
|- ( E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) -> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) |
82 |
81
|
rexlimivw |
|- ( E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) -> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) |
83 |
82
|
a1i13 |
|- ( ( # ` F ) = 2 -> ( E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) -> ( G e. USGraph -> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) ) |
84 |
|
fzo0to2pr |
|- ( 0 ..^ 2 ) = { 0 , 1 } |
85 |
10 84
|
eqtrdi |
|- ( ( # ` F ) = 2 -> ( 0 ..^ ( # ` F ) ) = { 0 , 1 } ) |
86 |
85
|
raleqdv |
|- ( ( # ` F ) = 2 -> ( A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> A. i e. { 0 , 1 } ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) |
87 |
|
2wlklem |
|- ( A. i e. { 0 , 1 } ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) |
88 |
86 87
|
bitrdi |
|- ( ( # ` F ) = 2 -> ( A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) |
89 |
88
|
imbi2d |
|- ( ( # ` F ) = 2 -> ( ( G e. USGraph -> A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) <-> ( G e. USGraph -> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( I ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) ) |
90 |
83 89
|
sylibrd |
|- ( ( # ` F ) = 2 -> ( E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) -> ( G e. USGraph -> A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) ) |
91 |
90
|
ad2antrr |
|- ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) -> ( E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) -> ( G e. USGraph -> A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) ) |
92 |
91
|
imp |
|- ( ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) -> ( G e. USGraph -> A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) |
93 |
92
|
imp |
|- ( ( ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) /\ G e. USGraph ) -> A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) |
94 |
51 59 93
|
3jca |
|- ( ( ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) /\ G e. USGraph ) -> ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) |
95 |
20
|
simprbi |
|- ( P : ( 0 ... 2 ) -1-1-> V -> Fun `' P ) |
96 |
95
|
adantl |
|- ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) -> Fun `' P ) |
97 |
96
|
ad2antrr |
|- ( ( ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) /\ G e. USGraph ) -> Fun `' P ) |
98 |
94 97
|
jca |
|- ( ( ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) /\ G e. USGraph ) -> ( ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ Fun `' P ) ) |
99 |
7 9
|
bitrd |
|- ( G e. UPGraph -> ( F ( SPaths ` G ) P <-> ( ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ Fun `' P ) ) ) |
100 |
4 99
|
syl |
|- ( G e. USGraph -> ( F ( SPaths ` G ) P <-> ( ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ Fun `' P ) ) ) |
101 |
100
|
adantl |
|- ( ( ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) /\ G e. USGraph ) -> ( F ( SPaths ` G ) P <-> ( ( F : ( 0 ..^ ( # ` F ) ) -1-1-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. i e. ( 0 ..^ ( # ` F ) ) ( I ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) /\ Fun `' P ) ) ) |
102 |
98 101
|
mpbird |
|- ( ( ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) /\ G e. USGraph ) -> F ( SPaths ` G ) P ) |
103 |
|
simpr |
|- ( ( ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) /\ G e. USGraph ) -> G e. USGraph ) |
104 |
|
simp-4l |
|- ( ( ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) /\ G e. USGraph ) -> ( # ` F ) = 2 ) |
105 |
103 104 3
|
syl2anc |
|- ( ( ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) /\ G e. USGraph ) -> ( F ( Paths ` G ) P <-> F ( SPaths ` G ) P ) ) |
106 |
102 105
|
mpbird |
|- ( ( ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) /\ G e. USGraph ) -> F ( Paths ` G ) P ) |
107 |
106 104
|
jca |
|- ( ( ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) /\ G e. USGraph ) -> ( F ( Paths ` G ) P /\ ( # ` F ) = 2 ) ) |
108 |
107
|
ex |
|- ( ( ( ( ( # ` F ) = 2 /\ F : ( 0 ..^ 2 ) -1-1-> dom I ) /\ P : ( 0 ... 2 ) -1-1-> V ) /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) -> ( G e. USGraph -> ( F ( Paths ` G ) P /\ ( # ` F ) = 2 ) ) ) |
109 |
108
|
exp41 |
|- ( ( # ` F ) = 2 -> ( F : ( 0 ..^ 2 ) -1-1-> dom I -> ( P : ( 0 ... 2 ) -1-1-> V -> ( E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) -> ( G e. USGraph -> ( F ( Paths ` G ) P /\ ( # ` F ) = 2 ) ) ) ) ) ) |
110 |
43 109
|
mpcom |
|- ( F : ( 0 ..^ 2 ) -1-1-> dom I -> ( P : ( 0 ... 2 ) -1-1-> V -> ( E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) -> ( G e. USGraph -> ( F ( Paths ` G ) P /\ ( # ` F ) = 2 ) ) ) ) ) |
111 |
110
|
3imp |
|- ( ( F : ( 0 ..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) -> ( G e. USGraph -> ( F ( Paths ` G ) P /\ ( # ` F ) = 2 ) ) ) |
112 |
111
|
com12 |
|- ( G e. USGraph -> ( ( F : ( 0 ..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) -> ( F ( Paths ` G ) P /\ ( # ` F ) = 2 ) ) ) |
113 |
39 112
|
impbid |
|- ( G e. USGraph -> ( ( F ( Paths ` G ) P /\ ( # ` F ) = 2 ) <-> ( F : ( 0 ..^ 2 ) -1-1-> dom I /\ P : ( 0 ... 2 ) -1-1-> V /\ E. x e. V E. y e. ( V \ { x } ) E. z e. ( V \ { x , y } ) ( ( ( P ` 0 ) = x /\ ( P ` 1 ) = y /\ ( P ` 2 ) = z ) /\ ( ( I ` ( F ` 0 ) ) = { x , y } /\ ( I ` ( F ` 1 ) ) = { y , z } ) ) ) ) ) |