Step |
Hyp |
Ref |
Expression |
1 |
|
usgrexmpl.v |
⊢ 𝑉 = ( 0 ... 4 ) |
2 |
|
usgrexmpl.e |
⊢ 𝐸 = 〈“ { 0 , 1 } { 1 , 2 } { 2 , 0 } { 0 , 3 } ”〉 |
3 |
|
usgrexmpl.g |
⊢ 𝐺 = 〈 𝑉 , 𝐸 〉 |
4 |
1 2 3
|
usgrexmpllem |
⊢ ( ( Vtx ‘ 𝐺 ) = 𝑉 ∧ ( iEdg ‘ 𝐺 ) = 𝐸 ) |
5 |
|
id |
⊢ ( ( Vtx ‘ 𝐺 ) = 𝑉 → ( Vtx ‘ 𝐺 ) = 𝑉 ) |
6 |
|
fz0to4untppr |
⊢ ( 0 ... 4 ) = ( { 0 , 1 , 2 } ∪ { 3 , 4 } ) |
7 |
1 6
|
eqtri |
⊢ 𝑉 = ( { 0 , 1 , 2 } ∪ { 3 , 4 } ) |
8 |
5 7
|
eqtrdi |
⊢ ( ( Vtx ‘ 𝐺 ) = 𝑉 → ( Vtx ‘ 𝐺 ) = ( { 0 , 1 , 2 } ∪ { 3 , 4 } ) ) |
9 |
8
|
adantr |
⊢ ( ( ( Vtx ‘ 𝐺 ) = 𝑉 ∧ ( iEdg ‘ 𝐺 ) = 𝐸 ) → ( Vtx ‘ 𝐺 ) = ( { 0 , 1 , 2 } ∪ { 3 , 4 } ) ) |
10 |
4 9
|
ax-mp |
⊢ ( Vtx ‘ 𝐺 ) = ( { 0 , 1 , 2 } ∪ { 3 , 4 } ) |