Step |
Hyp |
Ref |
Expression |
1 |
|
hashnbusgrvd.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
1
|
uvtxisvtx |
⊢ ( 𝑛 ∈ ( UnivVtx ‘ 𝐺 ) → 𝑛 ∈ 𝑉 ) |
3 |
|
fveqeq2 |
⊢ ( 𝑣 = 𝑛 → ( ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ↔ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑛 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) ) |
4 |
3
|
rspccv |
⊢ ( ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) → ( 𝑛 ∈ 𝑉 → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑛 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) ) |
5 |
4
|
adantl |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) → ( 𝑛 ∈ 𝑉 → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑛 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) ) |
6 |
5
|
imp |
⊢ ( ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) ∧ 𝑛 ∈ 𝑉 ) → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑛 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) |
7 |
1
|
usgruvtxvdb |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑛 ∈ 𝑉 ) → ( 𝑛 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑛 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) ) |
8 |
7
|
adantlr |
⊢ ( ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) ∧ 𝑛 ∈ 𝑉 ) → ( 𝑛 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑛 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) ) |
9 |
6 8
|
mpbird |
⊢ ( ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) ∧ 𝑛 ∈ 𝑉 ) → 𝑛 ∈ ( UnivVtx ‘ 𝐺 ) ) |
10 |
9
|
ex |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) → ( 𝑛 ∈ 𝑉 → 𝑛 ∈ ( UnivVtx ‘ 𝐺 ) ) ) |
11 |
2 10
|
impbid2 |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) → ( 𝑛 ∈ ( UnivVtx ‘ 𝐺 ) ↔ 𝑛 ∈ 𝑉 ) ) |
12 |
11
|
eqrdv |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) → ( UnivVtx ‘ 𝐺 ) = 𝑉 ) |
13 |
|
fusgrusgr |
⊢ ( 𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph ) |
14 |
1
|
cusgruvtxb |
⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ ComplUSGraph ↔ ( UnivVtx ‘ 𝐺 ) = 𝑉 ) ) |
15 |
13 14
|
syl |
⊢ ( 𝐺 ∈ FinUSGraph → ( 𝐺 ∈ ComplUSGraph ↔ ( UnivVtx ‘ 𝐺 ) = 𝑉 ) ) |
16 |
15
|
adantr |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) → ( 𝐺 ∈ ComplUSGraph ↔ ( UnivVtx ‘ 𝐺 ) = 𝑉 ) ) |
17 |
12 16
|
mpbird |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) → 𝐺 ∈ ComplUSGraph ) |
18 |
17
|
ex |
⊢ ( 𝐺 ∈ FinUSGraph → ( ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) → 𝐺 ∈ ComplUSGraph ) ) |