Step |
Hyp |
Ref |
Expression |
1 |
|
wlkp1.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
wlkp1.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
3 |
|
wlkp1.f |
⊢ ( 𝜑 → Fun 𝐼 ) |
4 |
|
wlkp1.a |
⊢ ( 𝜑 → 𝐼 ∈ Fin ) |
5 |
|
wlkp1.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
6 |
|
wlkp1.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑉 ) |
7 |
|
wlkp1.d |
⊢ ( 𝜑 → ¬ 𝐵 ∈ dom 𝐼 ) |
8 |
|
wlkp1.w |
⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) |
9 |
|
wlkp1.n |
⊢ 𝑁 = ( ♯ ‘ 𝐹 ) |
10 |
|
wlkp1.e |
⊢ ( 𝜑 → 𝐸 ∈ ( Edg ‘ 𝐺 ) ) |
11 |
|
wlkp1.x |
⊢ ( 𝜑 → { ( 𝑃 ‘ 𝑁 ) , 𝐶 } ⊆ 𝐸 ) |
12 |
|
wlkp1.u |
⊢ ( 𝜑 → ( iEdg ‘ 𝑆 ) = ( 𝐼 ∪ { 〈 𝐵 , 𝐸 〉 } ) ) |
13 |
|
wlkp1.h |
⊢ 𝐻 = ( 𝐹 ∪ { 〈 𝑁 , 𝐵 〉 } ) |
14 |
|
wlkp1.q |
⊢ 𝑄 = ( 𝑃 ∪ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ) |
15 |
|
wlkp1.s |
⊢ ( 𝜑 → ( Vtx ‘ 𝑆 ) = 𝑉 ) |
16 |
|
eqid |
⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) |
17 |
16
|
wlkf |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) |
18 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
19 |
18
|
wlkp |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) |
20 |
17 19
|
jca |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ) |
21 |
8 20
|
syl |
⊢ ( 𝜑 → ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ) |
22 |
6 15
|
eleqtrrd |
⊢ ( 𝜑 → 𝐶 ∈ ( Vtx ‘ 𝑆 ) ) |
23 |
22
|
elfvexd |
⊢ ( 𝜑 → 𝑆 ∈ V ) |
24 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ) → 𝑆 ∈ V ) |
25 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ) → 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) |
26 |
|
snex |
⊢ { 〈 𝑁 , 𝐵 〉 } ∈ V |
27 |
|
unexg |
⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ { 〈 𝑁 , 𝐵 〉 } ∈ V ) → ( 𝐹 ∪ { 〈 𝑁 , 𝐵 〉 } ) ∈ V ) |
28 |
25 26 27
|
sylancl |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ) → ( 𝐹 ∪ { 〈 𝑁 , 𝐵 〉 } ) ∈ V ) |
29 |
13 28
|
eqeltrid |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ) → 𝐻 ∈ V ) |
30 |
|
ovex |
⊢ ( 0 ... ( ♯ ‘ 𝐹 ) ) ∈ V |
31 |
|
fvex |
⊢ ( Vtx ‘ 𝐺 ) ∈ V |
32 |
30 31
|
fpm |
⊢ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → 𝑃 ∈ ( ( Vtx ‘ 𝐺 ) ↑pm ( 0 ... ( ♯ ‘ 𝐹 ) ) ) ) |
33 |
32
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ) → 𝑃 ∈ ( ( Vtx ‘ 𝐺 ) ↑pm ( 0 ... ( ♯ ‘ 𝐹 ) ) ) ) |
34 |
|
snex |
⊢ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ∈ V |
35 |
|
unexg |
⊢ ( ( 𝑃 ∈ ( ( Vtx ‘ 𝐺 ) ↑pm ( 0 ... ( ♯ ‘ 𝐹 ) ) ) ∧ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ∈ V ) → ( 𝑃 ∪ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ) ∈ V ) |
36 |
33 34 35
|
sylancl |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ) → ( 𝑃 ∪ { 〈 ( 𝑁 + 1 ) , 𝐶 〉 } ) ∈ V ) |
37 |
14 36
|
eqeltrid |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ) → 𝑄 ∈ V ) |
38 |
24 29 37
|
3jca |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ) → ( 𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V ) ) |
39 |
21 38
|
mpdan |
⊢ ( 𝜑 → ( 𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V ) ) |