Step |
Hyp |
Ref |
Expression |
1 |
|
cycliswlk |
⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) |
2 |
|
wlkv |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) |
3 |
1 2
|
syl |
⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) |
4 |
3
|
simp2d |
⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 ∈ V ) |
5 |
4
|
adantl |
⊢ ( ( 𝐺 ∈ AcyclicGraph ∧ 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ) → 𝐹 ∈ V ) |
6 |
3
|
simp3d |
⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝑃 ∈ V ) |
7 |
6
|
adantl |
⊢ ( ( 𝐺 ∈ AcyclicGraph ∧ 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ) → 𝑃 ∈ V ) |
8 |
|
breq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ↔ 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 ) ) |
9 |
|
eqeq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 = ∅ ↔ 𝐹 = ∅ ) ) |
10 |
8 9
|
imbi12d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 → 𝑓 = ∅ ) ↔ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 → 𝐹 = ∅ ) ) ) |
11 |
|
breq2 |
⊢ ( 𝑝 = 𝑃 → ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 ↔ 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ) ) |
12 |
11
|
imbi1d |
⊢ ( 𝑝 = 𝑃 → ( ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑝 → 𝐹 = ∅ ) ↔ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 = ∅ ) ) ) |
13 |
10 12
|
sylan9bb |
⊢ ( ( 𝑓 = 𝐹 ∧ 𝑝 = 𝑃 ) → ( ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 → 𝑓 = ∅ ) ↔ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 = ∅ ) ) ) |
14 |
|
isacycgr1 |
⊢ ( 𝐺 ∈ AcyclicGraph → ( 𝐺 ∈ AcyclicGraph ↔ ∀ 𝑓 ∀ 𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 → 𝑓 = ∅ ) ) ) |
15 |
14
|
ibi |
⊢ ( 𝐺 ∈ AcyclicGraph → ∀ 𝑓 ∀ 𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 → 𝑓 = ∅ ) ) |
16 |
15
|
19.21bbi |
⊢ ( 𝐺 ∈ AcyclicGraph → ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 → 𝑓 = ∅ ) ) |
17 |
16
|
adantr |
⊢ ( ( 𝐺 ∈ AcyclicGraph ∧ 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ) → ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 → 𝑓 = ∅ ) ) |
18 |
5 7 13 17
|
vtocl2d |
⊢ ( ( 𝐺 ∈ AcyclicGraph ∧ 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ) → ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 = ∅ ) ) |
19 |
18
|
ex |
⊢ ( 𝐺 ∈ AcyclicGraph → ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 = ∅ ) ) ) |
20 |
19
|
pm2.43d |
⊢ ( 𝐺 ∈ AcyclicGraph → ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → 𝐹 = ∅ ) ) |
21 |
20
|
imp |
⊢ ( ( 𝐺 ∈ AcyclicGraph ∧ 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ) → 𝐹 = ∅ ) |