Step |
Hyp |
Ref |
Expression |
1 |
|
vtxdushgrfvedg.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
vtxdushgrfvedg.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
3 |
|
vtxdushgrfvedg.d |
⊢ 𝐷 = ( VtxDeg ‘ 𝐺 ) |
4 |
3
|
fveq1i |
⊢ ( 𝐷 ‘ 𝑈 ) = ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑈 ) |
5 |
4
|
a1i |
⊢ ( ( 𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉 ) → ( 𝐷 ‘ 𝑈 ) = ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑈 ) ) |
6 |
|
eqid |
⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) |
7 |
|
eqid |
⊢ dom ( iEdg ‘ 𝐺 ) = dom ( iEdg ‘ 𝐺 ) |
8 |
1 6 7
|
vtxdgval |
⊢ ( 𝑈 ∈ 𝑉 → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑈 ) = ( ( ♯ ‘ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑈 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) } ) +𝑒 ( ♯ ‘ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) = { 𝑈 } } ) ) ) |
9 |
8
|
adantl |
⊢ ( ( 𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉 ) → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑈 ) = ( ( ♯ ‘ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑈 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) } ) +𝑒 ( ♯ ‘ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) = { 𝑈 } } ) ) ) |
10 |
1 2
|
vtxdushgrfvedglem |
⊢ ( ( 𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉 ) → ( ♯ ‘ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑈 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) } ) = ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒 } ) ) |
11 |
|
fvex |
⊢ ( iEdg ‘ 𝐺 ) ∈ V |
12 |
11
|
dmex |
⊢ dom ( iEdg ‘ 𝐺 ) ∈ V |
13 |
12
|
rabex |
⊢ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) = { 𝑈 } } ∈ V |
14 |
13
|
a1i |
⊢ ( ( 𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉 ) → { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) = { 𝑈 } } ∈ V ) |
15 |
|
eqid |
⊢ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) = { 𝑈 } } = { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) = { 𝑈 } } |
16 |
|
eqeq1 |
⊢ ( 𝑒 = 𝑐 → ( 𝑒 = { 𝑈 } ↔ 𝑐 = { 𝑈 } ) ) |
17 |
16
|
cbvrabv |
⊢ { 𝑒 ∈ 𝐸 ∣ 𝑒 = { 𝑈 } } = { 𝑐 ∈ 𝐸 ∣ 𝑐 = { 𝑈 } } |
18 |
|
eqid |
⊢ ( 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) = { 𝑈 } } ↦ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) = ( 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) = { 𝑈 } } ↦ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) |
19 |
2 6 15 17 18
|
ushgredgedgloop |
⊢ ( ( 𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉 ) → ( 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) = { 𝑈 } } ↦ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) : { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) = { 𝑈 } } –1-1-onto→ { 𝑒 ∈ 𝐸 ∣ 𝑒 = { 𝑈 } } ) |
20 |
14 19
|
hasheqf1od |
⊢ ( ( 𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉 ) → ( ♯ ‘ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) = { 𝑈 } } ) = ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑒 = { 𝑈 } } ) ) |
21 |
10 20
|
oveq12d |
⊢ ( ( 𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉 ) → ( ( ♯ ‘ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑈 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) } ) +𝑒 ( ♯ ‘ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) = { 𝑈 } } ) ) = ( ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒 } ) +𝑒 ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑒 = { 𝑈 } } ) ) ) |
22 |
5 9 21
|
3eqtrd |
⊢ ( ( 𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉 ) → ( 𝐷 ‘ 𝑈 ) = ( ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒 } ) +𝑒 ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑒 = { 𝑈 } } ) ) ) |