Step |
Hyp |
Ref |
Expression |
1 |
|
frgrncvvdeq.v1 |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
frgrncvvdeq.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
3 |
|
frgrncvvdeq.nx |
⊢ 𝐷 = ( 𝐺 NeighbVtx 𝑋 ) |
4 |
|
frgrncvvdeq.ny |
⊢ 𝑁 = ( 𝐺 NeighbVtx 𝑌 ) |
5 |
|
frgrncvvdeq.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
6 |
|
frgrncvvdeq.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
7 |
|
frgrncvvdeq.ne |
⊢ ( 𝜑 → 𝑋 ≠ 𝑌 ) |
8 |
|
frgrncvvdeq.xy |
⊢ ( 𝜑 → 𝑌 ∉ 𝐷 ) |
9 |
|
frgrncvvdeq.f |
⊢ ( 𝜑 → 𝐺 ∈ FriendGraph ) |
10 |
|
frgrncvvdeq.a |
⊢ 𝐴 = ( 𝑥 ∈ 𝐷 ↦ ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) ) |
11 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → 𝑥 ∈ 𝐷 ) |
12 |
|
riotaex |
⊢ ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) ∈ V |
13 |
10
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ 𝐷 ∧ ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) ∈ V ) → ( 𝐴 ‘ 𝑥 ) = ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) ) |
14 |
11 12 13
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ( 𝐴 ‘ 𝑥 ) = ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) ) |
15 |
14
|
sneqd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → { ( 𝐴 ‘ 𝑥 ) } = { ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) } ) |
16 |
1 2 3 4 5 6 7 8 9 10
|
frgrncvvdeqlem3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → { ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) } = ( ( 𝐺 NeighbVtx 𝑥 ) ∩ 𝑁 ) ) |
17 |
15 16
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → { ( 𝐴 ‘ 𝑥 ) } = ( ( 𝐺 NeighbVtx 𝑥 ) ∩ 𝑁 ) ) |