Step |
Hyp |
Ref |
Expression |
1 |
|
frgrwopreg.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
frgrwopreg.d |
⊢ 𝐷 = ( VtxDeg ‘ 𝐺 ) |
3 |
|
frgrwopreg.a |
⊢ 𝐴 = { 𝑥 ∈ 𝑉 ∣ ( 𝐷 ‘ 𝑥 ) = 𝐾 } |
4 |
|
frgrwopreg.b |
⊢ 𝐵 = ( 𝑉 ∖ 𝐴 ) |
5 |
|
frgrwopreg.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
6 |
1 2 3 4 5
|
frgrwopreglem4 |
⊢ ( 𝐺 ∈ FriendGraph → ∀ 𝑣 ∈ 𝐴 ∀ 𝑤 ∈ 𝐵 { 𝑣 , 𝑤 } ∈ 𝐸 ) |
7 |
|
snidg |
⊢ ( 𝑋 ∈ 𝑉 → 𝑋 ∈ { 𝑋 } ) |
8 |
7
|
adantr |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐴 = { 𝑋 } ) → 𝑋 ∈ { 𝑋 } ) |
9 |
|
eleq2 |
⊢ ( 𝐴 = { 𝑋 } → ( 𝑋 ∈ 𝐴 ↔ 𝑋 ∈ { 𝑋 } ) ) |
10 |
9
|
adantl |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐴 = { 𝑋 } ) → ( 𝑋 ∈ 𝐴 ↔ 𝑋 ∈ { 𝑋 } ) ) |
11 |
8 10
|
mpbird |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐴 = { 𝑋 } ) → 𝑋 ∈ 𝐴 ) |
12 |
|
preq1 |
⊢ ( 𝑣 = 𝑋 → { 𝑣 , 𝑤 } = { 𝑋 , 𝑤 } ) |
13 |
12
|
eleq1d |
⊢ ( 𝑣 = 𝑋 → ( { 𝑣 , 𝑤 } ∈ 𝐸 ↔ { 𝑋 , 𝑤 } ∈ 𝐸 ) ) |
14 |
13
|
ralbidv |
⊢ ( 𝑣 = 𝑋 → ( ∀ 𝑤 ∈ 𝐵 { 𝑣 , 𝑤 } ∈ 𝐸 ↔ ∀ 𝑤 ∈ 𝐵 { 𝑋 , 𝑤 } ∈ 𝐸 ) ) |
15 |
14
|
rspcv |
⊢ ( 𝑋 ∈ 𝐴 → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑤 ∈ 𝐵 { 𝑣 , 𝑤 } ∈ 𝐸 → ∀ 𝑤 ∈ 𝐵 { 𝑋 , 𝑤 } ∈ 𝐸 ) ) |
16 |
11 15
|
syl |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐴 = { 𝑋 } ) → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑤 ∈ 𝐵 { 𝑣 , 𝑤 } ∈ 𝐸 → ∀ 𝑤 ∈ 𝐵 { 𝑋 , 𝑤 } ∈ 𝐸 ) ) |
17 |
|
difeq2 |
⊢ ( 𝐴 = { 𝑋 } → ( 𝑉 ∖ 𝐴 ) = ( 𝑉 ∖ { 𝑋 } ) ) |
18 |
4 17
|
syl5eq |
⊢ ( 𝐴 = { 𝑋 } → 𝐵 = ( 𝑉 ∖ { 𝑋 } ) ) |
19 |
18
|
adantl |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐴 = { 𝑋 } ) → 𝐵 = ( 𝑉 ∖ { 𝑋 } ) ) |
20 |
19
|
raleqdv |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐴 = { 𝑋 } ) → ( ∀ 𝑤 ∈ 𝐵 { 𝑋 , 𝑤 } ∈ 𝐸 ↔ ∀ 𝑤 ∈ ( 𝑉 ∖ { 𝑋 } ) { 𝑋 , 𝑤 } ∈ 𝐸 ) ) |
21 |
16 20
|
sylibd |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐴 = { 𝑋 } ) → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑤 ∈ 𝐵 { 𝑣 , 𝑤 } ∈ 𝐸 → ∀ 𝑤 ∈ ( 𝑉 ∖ { 𝑋 } ) { 𝑋 , 𝑤 } ∈ 𝐸 ) ) |
22 |
6 21
|
syl5com |
⊢ ( 𝐺 ∈ FriendGraph → ( ( 𝑋 ∈ 𝑉 ∧ 𝐴 = { 𝑋 } ) → ∀ 𝑤 ∈ ( 𝑉 ∖ { 𝑋 } ) { 𝑋 , 𝑤 } ∈ 𝐸 ) ) |
23 |
22
|
3impib |
⊢ ( ( 𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝐴 = { 𝑋 } ) → ∀ 𝑤 ∈ ( 𝑉 ∖ { 𝑋 } ) { 𝑋 , 𝑤 } ∈ 𝐸 ) |