Step |
Hyp |
Ref |
Expression |
1 |
|
eupth0.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
eupth0.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
3 |
|
eupthres.d |
⊢ ( 𝜑 → 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 ) |
4 |
|
eupthres.n |
⊢ ( 𝜑 → 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
5 |
|
eupthres.e |
⊢ ( 𝜑 → ( iEdg ‘ 𝑆 ) = ( 𝐼 ↾ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ) ) |
6 |
|
eupthres.h |
⊢ 𝐻 = ( 𝐹 prefix 𝑁 ) |
7 |
|
eupthres.q |
⊢ 𝑄 = ( 𝑃 ↾ ( 0 ... 𝑁 ) ) |
8 |
|
eupthres.s |
⊢ ( Vtx ‘ 𝑆 ) = 𝑉 |
9 |
|
eupthistrl |
⊢ ( 𝐹 ( EulerPaths ‘ 𝐺 ) 𝑃 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ) |
10 |
|
trliswlk |
⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) |
11 |
3 9 10
|
3syl |
⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) |
12 |
8
|
a1i |
⊢ ( 𝜑 → ( Vtx ‘ 𝑆 ) = 𝑉 ) |
13 |
1 2 11 4 12 5 6 7
|
wlkres |
⊢ ( 𝜑 → 𝐻 ( Walks ‘ 𝑆 ) 𝑄 ) |
14 |
3 9
|
syl |
⊢ ( 𝜑 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ) |
15 |
1 2 14 4 6
|
trlreslem |
⊢ ( 𝜑 → 𝐻 : ( 0 ..^ ( ♯ ‘ 𝐻 ) ) –1-1-onto→ dom ( 𝐼 ↾ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ) ) |
16 |
|
eqid |
⊢ ( iEdg ‘ 𝑆 ) = ( iEdg ‘ 𝑆 ) |
17 |
16
|
iseupthf1o |
⊢ ( 𝐻 ( EulerPaths ‘ 𝑆 ) 𝑄 ↔ ( 𝐻 ( Walks ‘ 𝑆 ) 𝑄 ∧ 𝐻 : ( 0 ..^ ( ♯ ‘ 𝐻 ) ) –1-1-onto→ dom ( iEdg ‘ 𝑆 ) ) ) |
18 |
5
|
dmeqd |
⊢ ( 𝜑 → dom ( iEdg ‘ 𝑆 ) = dom ( 𝐼 ↾ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ) ) |
19 |
18
|
f1oeq3d |
⊢ ( 𝜑 → ( 𝐻 : ( 0 ..^ ( ♯ ‘ 𝐻 ) ) –1-1-onto→ dom ( iEdg ‘ 𝑆 ) ↔ 𝐻 : ( 0 ..^ ( ♯ ‘ 𝐻 ) ) –1-1-onto→ dom ( 𝐼 ↾ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ) ) ) |
20 |
19
|
anbi2d |
⊢ ( 𝜑 → ( ( 𝐻 ( Walks ‘ 𝑆 ) 𝑄 ∧ 𝐻 : ( 0 ..^ ( ♯ ‘ 𝐻 ) ) –1-1-onto→ dom ( iEdg ‘ 𝑆 ) ) ↔ ( 𝐻 ( Walks ‘ 𝑆 ) 𝑄 ∧ 𝐻 : ( 0 ..^ ( ♯ ‘ 𝐻 ) ) –1-1-onto→ dom ( 𝐼 ↾ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ) ) ) ) |
21 |
17 20
|
syl5bb |
⊢ ( 𝜑 → ( 𝐻 ( EulerPaths ‘ 𝑆 ) 𝑄 ↔ ( 𝐻 ( Walks ‘ 𝑆 ) 𝑄 ∧ 𝐻 : ( 0 ..^ ( ♯ ‘ 𝐻 ) ) –1-1-onto→ dom ( 𝐼 ↾ ( 𝐹 “ ( 0 ..^ 𝑁 ) ) ) ) ) ) |
22 |
13 15 21
|
mpbir2and |
⊢ ( 𝜑 → 𝐻 ( EulerPaths ‘ 𝑆 ) 𝑄 ) |