Step |
Hyp |
Ref |
Expression |
1 |
|
hashnbusgrvd.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
fusgrusgr |
⊢ ( 𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph ) |
3 |
1
|
cusgruvtxb |
⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ ComplUSGraph ↔ ( UnivVtx ‘ 𝐺 ) = 𝑉 ) ) |
4 |
1
|
uvtxssvtx |
⊢ ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉 |
5 |
|
eqcom |
⊢ ( ( UnivVtx ‘ 𝐺 ) = 𝑉 ↔ 𝑉 = ( UnivVtx ‘ 𝐺 ) ) |
6 |
|
sssseq |
⊢ ( ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉 → ( 𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ↔ 𝑉 = ( UnivVtx ‘ 𝐺 ) ) ) |
7 |
5 6
|
bitr4id |
⊢ ( ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉 → ( ( UnivVtx ‘ 𝐺 ) = 𝑉 ↔ 𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ) ) |
8 |
4 7
|
mp1i |
⊢ ( 𝐺 ∈ USGraph → ( ( UnivVtx ‘ 𝐺 ) = 𝑉 ↔ 𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ) ) |
9 |
3 8
|
bitrd |
⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ ComplUSGraph ↔ 𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ) ) |
10 |
|
dfss3 |
⊢ ( 𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) |
11 |
9 10
|
bitrdi |
⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ ComplUSGraph ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) ) |
12 |
2 11
|
syl |
⊢ ( 𝐺 ∈ FinUSGraph → ( 𝐺 ∈ ComplUSGraph ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) ) |
13 |
1
|
usgruvtxvdb |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉 ) → ( 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) ) |
14 |
13
|
ralbidva |
⊢ ( 𝐺 ∈ FinUSGraph → ( ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ↔ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) ) |
15 |
12 14
|
bitrd |
⊢ ( 𝐺 ∈ FinUSGraph → ( 𝐺 ∈ ComplUSGraph ↔ ∀ 𝑣 ∈ 𝑉 ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑣 ) = ( ( ♯ ‘ 𝑉 ) − 1 ) ) ) |