Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
2 |
|
eqid |
⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) |
3 |
1 2
|
usgrfs |
⊢ ( 𝐺 ∈ USGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) |
4 |
|
fvex |
⊢ ( Vtx ‘ 𝐺 ) ∈ V |
5 |
|
fvex |
⊢ ( iEdg ‘ 𝐺 ) ∈ V |
6 |
4 5
|
pm3.2i |
⊢ ( ( Vtx ‘ 𝐺 ) ∈ V ∧ ( iEdg ‘ 𝐺 ) ∈ V ) |
7 |
|
isusgrop |
⊢ ( ( ( Vtx ‘ 𝐺 ) ∈ V ∧ ( iEdg ‘ 𝐺 ) ∈ V ) → ( 〈 ( Vtx ‘ 𝐺 ) , ( iEdg ‘ 𝐺 ) 〉 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) ) |
8 |
6 7
|
mp1i |
⊢ ( 𝐺 ∈ USGraph → ( 〈 ( Vtx ‘ 𝐺 ) , ( iEdg ‘ 𝐺 ) 〉 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) ) |
9 |
3 8
|
mpbird |
⊢ ( 𝐺 ∈ USGraph → 〈 ( Vtx ‘ 𝐺 ) , ( iEdg ‘ 𝐺 ) 〉 ∈ USGraph ) |