Step |
Hyp |
Ref |
Expression |
1 |
|
nbusgrf1o1.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
nbusgrf1o1.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
3 |
|
nbusgrf1o1.n |
⊢ 𝑁 = ( 𝐺 NeighbVtx 𝑈 ) |
4 |
|
nbusgrf1o1.i |
⊢ 𝐼 = { 𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒 } |
5 |
3
|
ovexi |
⊢ 𝑁 ∈ V |
6 |
|
mptexg |
⊢ ( 𝑁 ∈ V → ( 𝑛 ∈ 𝑁 ↦ { 𝑈 , 𝑛 } ) ∈ V ) |
7 |
5 6
|
mp1i |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉 ) → ( 𝑛 ∈ 𝑁 ↦ { 𝑈 , 𝑛 } ) ∈ V ) |
8 |
|
eqid |
⊢ ( 𝑛 ∈ 𝑁 ↦ { 𝑈 , 𝑛 } ) = ( 𝑛 ∈ 𝑁 ↦ { 𝑈 , 𝑛 } ) |
9 |
1 2 3 4 8
|
nbusgrf1o0 |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉 ) → ( 𝑛 ∈ 𝑁 ↦ { 𝑈 , 𝑛 } ) : 𝑁 –1-1-onto→ 𝐼 ) |
10 |
|
f1oeq1 |
⊢ ( 𝑓 = ( 𝑛 ∈ 𝑁 ↦ { 𝑈 , 𝑛 } ) → ( 𝑓 : 𝑁 –1-1-onto→ 𝐼 ↔ ( 𝑛 ∈ 𝑁 ↦ { 𝑈 , 𝑛 } ) : 𝑁 –1-1-onto→ 𝐼 ) ) |
11 |
7 9 10
|
spcedv |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉 ) → ∃ 𝑓 𝑓 : 𝑁 –1-1-onto→ 𝐼 ) |