Step |
Hyp |
Ref |
Expression |
1 |
|
ispth |
⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 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 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) → ( 𝐽 = 0 ∨ 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∨ 𝐽 = ( ♯ ‘ 𝐹 ) ) ) |
6 |
|
elfzo1 |
⊢ ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ↔ ( 𝐼 ∈ ℕ ∧ ( ♯ ‘ 𝐹 ) ∈ ℕ ∧ 𝐼 < ( ♯ ‘ 𝐹 ) ) ) |
7 |
|
nnnn0 |
⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( ♯ ‘ 𝐹 ) ∈ ℕ0 ) |
8 |
7
|
3ad2ant2 |
⊢ ( ( 𝐼 ∈ ℕ ∧ ( ♯ ‘ 𝐹 ) ∈ ℕ ∧ 𝐼 < ( ♯ ‘ 𝐹 ) ) → ( ♯ ‘ 𝐹 ) ∈ ℕ0 ) |
9 |
6 8
|
sylbi |
⊢ ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) → ( ♯ ‘ 𝐹 ) ∈ ℕ0 ) |
10 |
9
|
adantl |
⊢ ( ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ♯ ‘ 𝐹 ) ∈ ℕ0 ) |
11 |
|
fvinim0ffz |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐹 ) ∈ ℕ0 ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ↔ ( ( 𝑃 ‘ 0 ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ) |
12 |
10 11
|
sylan2 |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ↔ ( ( 𝑃 ‘ 0 ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∉ ( 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 ∈ ( 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 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → Fun 𝑃 ) |
18 |
|
fdm |
⊢ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → dom 𝑃 = ( 0 ... ( ♯ ‘ 𝐹 ) ) ) |
19 |
|
fzo0ss1 |
⊢ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ⊆ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) |
20 |
|
fzossfz |
⊢ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⊆ ( 0 ... ( ♯ ‘ 𝐹 ) ) |
21 |
19 20
|
sstri |
⊢ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ⊆ ( 0 ... ( ♯ ‘ 𝐹 ) ) |
22 |
21
|
sseli |
⊢ ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) → 𝐼 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ) |
23 |
|
eleq2 |
⊢ ( dom 𝑃 = ( 0 ... ( ♯ ‘ 𝐹 ) ) → ( 𝐼 ∈ dom 𝑃 ↔ 𝐼 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ) ) |
24 |
22 23
|
syl5ibr |
⊢ ( dom 𝑃 = ( 0 ... ( ♯ ‘ 𝐹 ) ) → ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) → 𝐼 ∈ dom 𝑃 ) ) |
25 |
18 24
|
syl |
⊢ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) → 𝐼 ∈ dom 𝑃 ) ) |
26 |
25
|
imp |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝐼 ∈ dom 𝑃 ) |
27 |
17 26
|
jca |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃 ) ) |
28 |
27
|
adantrl |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃 ) ) |
29 |
|
simprr |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) |
30 |
|
funfvima |
⊢ ( ( Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃 ) → ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝑃 ‘ 𝐼 ) ∈ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) |
31 |
28 29 30
|
sylc |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝑃 ‘ 𝐼 ) ∈ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
32 |
|
eleq1 |
⊢ ( ( 𝑃 ‘ 𝐼 ) = ( 𝑃 ‘ 0 ) → ( ( 𝑃 ‘ 𝐼 ) ∈ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ↔ ( 𝑃 ‘ 0 ) ∈ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) |
33 |
31 32
|
syl5ibcom |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑃 ‘ 𝐼 ) = ( 𝑃 ‘ 0 ) → ( 𝑃 ‘ 0 ) ∈ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) |
34 |
15 33
|
sylbid |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑃 ‘ 𝐼 ) = ( 𝑃 ‘ 𝐽 ) → ( 𝑃 ‘ 0 ) ∈ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) |
35 |
|
nnel |
⊢ ( ¬ ( 𝑃 ‘ 0 ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ↔ ( 𝑃 ‘ 0 ) ∈ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
36 |
34 35
|
syl6ibr |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑃 ‘ 𝐼 ) = ( 𝑃 ‘ 𝐽 ) → ¬ ( 𝑃 ‘ 0 ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) |
37 |
36
|
necon2ad |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑃 ‘ 0 ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) |
38 |
37
|
adantrd |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( ( 𝑃 ‘ 0 ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) |
39 |
12 38
|
sylbid |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) |
40 |
39
|
ex |
⊢ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) |
41 |
40
|
com23 |
⊢ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) |
42 |
41
|
a1d |
⊢ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) ) |
43 |
42
|
3imp |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) |
44 |
43
|
com12 |
⊢ ( ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) |
45 |
44
|
a1d |
⊢ ( ( 𝐽 = 0 ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐼 ≠ 𝐽 → ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) |
46 |
45
|
ex |
⊢ ( 𝐽 = 0 → ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐼 ≠ 𝐽 → ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) ) |
47 |
|
fvres |
⊢ ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) → ( ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ‘ 𝐼 ) = ( 𝑃 ‘ 𝐼 ) ) |
48 |
47
|
adantl |
⊢ ( ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ‘ 𝐼 ) = ( 𝑃 ‘ 𝐼 ) ) |
49 |
48
|
adantl |
⊢ ( ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ‘ 𝐼 ) = ( 𝑃 ‘ 𝐼 ) ) |
50 |
49
|
eqcomd |
⊢ ( ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝑃 ‘ 𝐼 ) = ( ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ‘ 𝐼 ) ) |
51 |
|
fvres |
⊢ ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) → ( ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ‘ 𝐽 ) = ( 𝑃 ‘ 𝐽 ) ) |
52 |
51
|
ad2antrl |
⊢ ( ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ‘ 𝐽 ) = ( 𝑃 ‘ 𝐽 ) ) |
53 |
52
|
eqcomd |
⊢ ( ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝑃 ‘ 𝐽 ) = ( ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ‘ 𝐽 ) ) |
54 |
50 53
|
eqeq12d |
⊢ ( ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑃 ‘ 𝐼 ) = ( 𝑃 ‘ 𝐽 ) ↔ ( ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ‘ 𝐼 ) = ( ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ‘ 𝐽 ) ) ) |
55 |
|
fssres |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ⊆ ( 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 ) } ) ∩ ( 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 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
62 |
61
|
ancomd |
⊢ ( ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
63 |
|
f1veqaeq |
⊢ ( ( ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) : ( 1 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ ( Vtx ‘ 𝐺 ) ∧ ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐽 ∈ ( 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 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ∈ ( 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 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑃 ‘ 𝐼 ) = ( 𝑃 ‘ 𝐽 ) → 𝐼 = 𝐽 ) ) |
66 |
65
|
ancoms |
⊢ ( ( ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ) → ( ( 𝑃 ‘ 𝐼 ) = ( 𝑃 ‘ 𝐽 ) → 𝐼 = 𝐽 ) ) |
67 |
66
|
necon3d |
⊢ ( ( ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ) → ( 𝐼 ≠ 𝐽 → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) |
68 |
67
|
ex |
⊢ ( ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝐼 ≠ 𝐽 → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) |
69 |
68
|
com23 |
⊢ ( ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐼 ≠ 𝐽 → ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) |
70 |
69
|
ex |
⊢ ( 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐼 ≠ 𝐽 → ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) ) |
71 |
9
|
adantl |
⊢ ( ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ♯ ‘ 𝐹 ) ∈ ℕ0 ) |
72 |
71 11
|
sylan2 |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ↔ ( ( 𝑃 ‘ 0 ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∉ ( 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 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑃 ‘ 𝐼 ) = ( 𝑃 ‘ 𝐽 ) ↔ ( 𝑃 ‘ 𝐼 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) |
76 |
27
|
adantrl |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃 ) ) |
77 |
|
simprr |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) |
78 |
76 77 30
|
sylc |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝑃 ‘ 𝐼 ) ∈ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
79 |
|
eleq1 |
⊢ ( ( 𝑃 ‘ 𝐼 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) → ( ( 𝑃 ‘ 𝐼 ) ∈ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ↔ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∈ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) |
80 |
78 79
|
syl5ibcom |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑃 ‘ 𝐼 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) → ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∈ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) |
81 |
75 80
|
sylbid |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑃 ‘ 𝐼 ) = ( 𝑃 ‘ 𝐽 ) → ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∈ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) |
82 |
|
nnel |
⊢ ( ¬ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ↔ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∈ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
83 |
81 82
|
syl6ibr |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑃 ‘ 𝐼 ) = ( 𝑃 ‘ 𝐽 ) → ¬ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) |
84 |
83
|
necon2ad |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) |
85 |
84
|
adantld |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( ( 𝑃 ‘ 0 ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) |
86 |
72 85
|
sylbid |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) |
87 |
86
|
ex |
⊢ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) |
88 |
87
|
com23 |
⊢ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) |
89 |
88
|
a1d |
⊢ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) ) |
90 |
89
|
3imp |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) |
91 |
90
|
com12 |
⊢ ( ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) |
92 |
91
|
a1d |
⊢ ( ( 𝐽 = ( ♯ ‘ 𝐹 ) ∧ 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐼 ≠ 𝐽 → ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) |
93 |
92
|
ex |
⊢ ( 𝐽 = ( ♯ ‘ 𝐹 ) → ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐼 ≠ 𝐽 → ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) ) |
94 |
46 70 93
|
3jaoi |
⊢ ( ( 𝐽 = 0 ∨ 𝐽 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∨ 𝐽 = ( ♯ ‘ 𝐹 ) ) → ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐼 ≠ 𝐽 → ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) ) |
95 |
5 94
|
syl |
⊢ ( 𝐽 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) → ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐼 ≠ 𝐽 → ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) ) |
96 |
95
|
3imp21 |
⊢ ( ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐽 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ≠ 𝐽 ) → ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) |
97 |
96
|
com12 |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐽 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ≠ 𝐽 ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) |
98 |
97
|
3exp |
⊢ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐽 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ≠ 𝐽 ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) ) |
99 |
2 4 98
|
3syl |
⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → ( Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐽 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ≠ 𝐽 ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) ) ) |
100 |
99
|
3imp |
⊢ ( ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐽 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ≠ 𝐽 ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) |
101 |
1 100
|
sylbi |
⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐽 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ≠ 𝐽 ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) ) |
102 |
101
|
imp |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 𝐼 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 𝐽 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∧ 𝐼 ≠ 𝐽 ) ) → ( 𝑃 ‘ 𝐼 ) ≠ ( 𝑃 ‘ 𝐽 ) ) |