Step |
Hyp |
Ref |
Expression |
1 |
|
uvtxel.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
isuvtx.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
3 |
1
|
uvtxval |
⊢ ( UnivVtx ‘ 𝐺 ) = { 𝑣 ∈ 𝑉 ∣ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑘 ∈ ( 𝐺 NeighbVtx 𝑣 ) } |
4 |
1 2
|
nbgrel |
⊢ ( 𝑘 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ( ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ∧ 𝑘 ≠ 𝑣 ∧ ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 ) ) |
5 |
|
df-3an |
⊢ ( ( ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ∧ 𝑘 ≠ 𝑣 ∧ ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 ) ↔ ( ( ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ∧ 𝑘 ≠ 𝑣 ) ∧ ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 ) ) |
6 |
4 5
|
bitri |
⊢ ( 𝑘 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ( ( ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ∧ 𝑘 ≠ 𝑣 ) ∧ ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 ) ) |
7 |
|
prcom |
⊢ { 𝑘 , 𝑣 } = { 𝑣 , 𝑘 } |
8 |
7
|
sseq1i |
⊢ ( { 𝑘 , 𝑣 } ⊆ 𝑒 ↔ { 𝑣 , 𝑘 } ⊆ 𝑒 ) |
9 |
8
|
rexbii |
⊢ ( ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑣 } ⊆ 𝑒 ↔ ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 ) |
10 |
|
id |
⊢ ( 𝑣 ∈ 𝑉 → 𝑣 ∈ 𝑉 ) |
11 |
|
eldifi |
⊢ ( 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) → 𝑘 ∈ 𝑉 ) |
12 |
10 11
|
anim12ci |
⊢ ( ( 𝑣 ∈ 𝑉 ∧ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ) → ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ) |
13 |
|
eldifsni |
⊢ ( 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) → 𝑘 ≠ 𝑣 ) |
14 |
13
|
adantl |
⊢ ( ( 𝑣 ∈ 𝑉 ∧ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ) → 𝑘 ≠ 𝑣 ) |
15 |
12 14
|
jca |
⊢ ( ( 𝑣 ∈ 𝑉 ∧ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ) → ( ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ∧ 𝑘 ≠ 𝑣 ) ) |
16 |
15
|
biantrurd |
⊢ ( ( 𝑣 ∈ 𝑉 ∧ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ) → ( ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 ↔ ( ( ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ∧ 𝑘 ≠ 𝑣 ) ∧ ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 ) ) ) |
17 |
9 16
|
bitr2id |
⊢ ( ( 𝑣 ∈ 𝑉 ∧ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ) → ( ( ( ( 𝑘 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉 ) ∧ 𝑘 ≠ 𝑣 ) ∧ ∃ 𝑒 ∈ 𝐸 { 𝑣 , 𝑘 } ⊆ 𝑒 ) ↔ ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑣 } ⊆ 𝑒 ) ) |
18 |
6 17
|
syl5bb |
⊢ ( ( 𝑣 ∈ 𝑉 ∧ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ) → ( 𝑘 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑣 } ⊆ 𝑒 ) ) |
19 |
18
|
ralbidva |
⊢ ( 𝑣 ∈ 𝑉 → ( ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑘 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑣 } ⊆ 𝑒 ) ) |
20 |
19
|
rabbiia |
⊢ { 𝑣 ∈ 𝑉 ∣ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑘 ∈ ( 𝐺 NeighbVtx 𝑣 ) } = { 𝑣 ∈ 𝑉 ∣ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑣 } ⊆ 𝑒 } |
21 |
3 20
|
eqtri |
⊢ ( UnivVtx ‘ 𝐺 ) = { 𝑣 ∈ 𝑉 ∣ ∀ 𝑘 ∈ ( 𝑉 ∖ { 𝑣 } ) ∃ 𝑒 ∈ 𝐸 { 𝑘 , 𝑣 } ⊆ 𝑒 } |