Step |
Hyp |
Ref |
Expression |
1 |
|
0clwlk.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
fz0sn |
⊢ ( 0 ... 0 ) = { 0 } |
3 |
2
|
eqcomi |
⊢ { 0 } = ( 0 ... 0 ) |
4 |
3
|
feq2i |
⊢ ( 𝑃 : { 0 } ⟶ { 𝑋 } ↔ 𝑃 : ( 0 ... 0 ) ⟶ { 𝑋 } ) |
5 |
4
|
biimpi |
⊢ ( 𝑃 : { 0 } ⟶ { 𝑋 } → 𝑃 : ( 0 ... 0 ) ⟶ { 𝑋 } ) |
6 |
5
|
3ad2ant3 |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 = ∅ ∧ 𝑃 : { 0 } ⟶ { 𝑋 } ) → 𝑃 : ( 0 ... 0 ) ⟶ { 𝑋 } ) |
7 |
|
snssi |
⊢ ( 𝑋 ∈ 𝑉 → { 𝑋 } ⊆ 𝑉 ) |
8 |
7
|
3ad2ant1 |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 = ∅ ∧ 𝑃 : { 0 } ⟶ { 𝑋 } ) → { 𝑋 } ⊆ 𝑉 ) |
9 |
6 8
|
fssd |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 = ∅ ∧ 𝑃 : { 0 } ⟶ { 𝑋 } ) → 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) |
10 |
|
breq1 |
⊢ ( 𝐹 = ∅ → ( 𝐹 ( ClWalks ‘ 𝐺 ) 𝑃 ↔ ∅ ( ClWalks ‘ 𝐺 ) 𝑃 ) ) |
11 |
10
|
3ad2ant2 |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 = ∅ ∧ 𝑃 : { 0 } ⟶ { 𝑋 } ) → ( 𝐹 ( ClWalks ‘ 𝐺 ) 𝑃 ↔ ∅ ( ClWalks ‘ 𝐺 ) 𝑃 ) ) |
12 |
1
|
1vgrex |
⊢ ( 𝑋 ∈ 𝑉 → 𝐺 ∈ V ) |
13 |
1
|
0clwlk |
⊢ ( 𝐺 ∈ V → ( ∅ ( ClWalks ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) ) |
14 |
12 13
|
syl |
⊢ ( 𝑋 ∈ 𝑉 → ( ∅ ( ClWalks ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) ) |
15 |
14
|
3ad2ant1 |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 = ∅ ∧ 𝑃 : { 0 } ⟶ { 𝑋 } ) → ( ∅ ( ClWalks ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) ) |
16 |
11 15
|
bitrd |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 = ∅ ∧ 𝑃 : { 0 } ⟶ { 𝑋 } ) → ( 𝐹 ( ClWalks ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) ) |
17 |
9 16
|
mpbird |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 = ∅ ∧ 𝑃 : { 0 } ⟶ { 𝑋 } ) → 𝐹 ( ClWalks ‘ 𝐺 ) 𝑃 ) |