Step |
Hyp |
Ref |
Expression |
1 |
|
vtxdgval.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
vtxdgval.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
3 |
|
vtxdgval.a |
⊢ 𝐴 = dom 𝐼 |
4 |
1
|
1vgrex |
⊢ ( 𝑈 ∈ 𝑉 → 𝐺 ∈ V ) |
5 |
1 2 3
|
vtxdgfval |
⊢ ( 𝐺 ∈ V → ( VtxDeg ‘ 𝐺 ) = ( 𝑢 ∈ 𝑉 ↦ ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑢 ∈ ( 𝐼 ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑢 } } ) ) ) ) |
6 |
4 5
|
syl |
⊢ ( 𝑈 ∈ 𝑉 → ( VtxDeg ‘ 𝐺 ) = ( 𝑢 ∈ 𝑉 ↦ ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑢 ∈ ( 𝐼 ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑢 } } ) ) ) ) |
7 |
6
|
fveq1d |
⊢ ( 𝑈 ∈ 𝑉 → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑈 ) = ( ( 𝑢 ∈ 𝑉 ↦ ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑢 ∈ ( 𝐼 ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑢 } } ) ) ) ‘ 𝑈 ) ) |
8 |
|
eleq1 |
⊢ ( 𝑢 = 𝑈 → ( 𝑢 ∈ ( 𝐼 ‘ 𝑥 ) ↔ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) ) ) |
9 |
8
|
rabbidv |
⊢ ( 𝑢 = 𝑈 → { 𝑥 ∈ 𝐴 ∣ 𝑢 ∈ ( 𝐼 ‘ 𝑥 ) } = { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) |
10 |
9
|
fveq2d |
⊢ ( 𝑢 = 𝑈 → ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑢 ∈ ( 𝐼 ‘ 𝑥 ) } ) = ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) ) |
11 |
|
sneq |
⊢ ( 𝑢 = 𝑈 → { 𝑢 } = { 𝑈 } ) |
12 |
11
|
eqeq2d |
⊢ ( 𝑢 = 𝑈 → ( ( 𝐼 ‘ 𝑥 ) = { 𝑢 } ↔ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } ) ) |
13 |
12
|
rabbidv |
⊢ ( 𝑢 = 𝑈 → { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑢 } } = { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) |
14 |
13
|
fveq2d |
⊢ ( 𝑢 = 𝑈 → ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑢 } } ) = ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ) |
15 |
10 14
|
oveq12d |
⊢ ( 𝑢 = 𝑈 → ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑢 ∈ ( 𝐼 ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑢 } } ) ) = ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ) ) |
16 |
|
eqid |
⊢ ( 𝑢 ∈ 𝑉 ↦ ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑢 ∈ ( 𝐼 ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑢 } } ) ) ) = ( 𝑢 ∈ 𝑉 ↦ ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑢 ∈ ( 𝐼 ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑢 } } ) ) ) |
17 |
|
ovex |
⊢ ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ) ∈ V |
18 |
15 16 17
|
fvmpt |
⊢ ( 𝑈 ∈ 𝑉 → ( ( 𝑢 ∈ 𝑉 ↦ ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑢 ∈ ( 𝐼 ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑢 } } ) ) ) ‘ 𝑈 ) = ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ) ) |
19 |
7 18
|
eqtrd |
⊢ ( 𝑈 ∈ 𝑉 → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑈 ) = ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ) ) |