Step |
Hyp |
Ref |
Expression |
1 |
|
vtxdeqd.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝑋 ) |
2 |
|
vtxdeqd.h |
⊢ ( 𝜑 → 𝐻 ∈ 𝑌 ) |
3 |
|
vtxdeqd.v |
⊢ ( 𝜑 → ( Vtx ‘ 𝐻 ) = ( Vtx ‘ 𝐺 ) ) |
4 |
|
vtxdeqd.i |
⊢ ( 𝜑 → ( iEdg ‘ 𝐻 ) = ( iEdg ‘ 𝐺 ) ) |
5 |
4
|
dmeqd |
⊢ ( 𝜑 → dom ( iEdg ‘ 𝐻 ) = dom ( iEdg ‘ 𝐺 ) ) |
6 |
4
|
fveq1d |
⊢ ( 𝜑 → ( ( iEdg ‘ 𝐻 ) ‘ 𝑥 ) = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) |
7 |
6
|
eleq2d |
⊢ ( 𝜑 → ( 𝑢 ∈ ( ( iEdg ‘ 𝐻 ) ‘ 𝑥 ) ↔ 𝑢 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) ) |
8 |
5 7
|
rabeqbidv |
⊢ ( 𝜑 → { 𝑥 ∈ dom ( iEdg ‘ 𝐻 ) ∣ 𝑢 ∈ ( ( iEdg ‘ 𝐻 ) ‘ 𝑥 ) } = { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑢 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) } ) |
9 |
8
|
fveq2d |
⊢ ( 𝜑 → ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐻 ) ∣ 𝑢 ∈ ( ( iEdg ‘ 𝐻 ) ‘ 𝑥 ) } ) = ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑢 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) } ) ) |
10 |
6
|
eqeq1d |
⊢ ( 𝜑 → ( ( ( iEdg ‘ 𝐻 ) ‘ 𝑥 ) = { 𝑢 } ↔ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) = { 𝑢 } ) ) |
11 |
5 10
|
rabeqbidv |
⊢ ( 𝜑 → { 𝑥 ∈ dom ( iEdg ‘ 𝐻 ) ∣ ( ( iEdg ‘ 𝐻 ) ‘ 𝑥 ) = { 𝑢 } } = { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) = { 𝑢 } } ) |
12 |
11
|
fveq2d |
⊢ ( 𝜑 → ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐻 ) ∣ ( ( iEdg ‘ 𝐻 ) ‘ 𝑥 ) = { 𝑢 } } ) = ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) = { 𝑢 } } ) ) |
13 |
9 12
|
oveq12d |
⊢ ( 𝜑 → ( ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐻 ) ∣ 𝑢 ∈ ( ( iEdg ‘ 𝐻 ) ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐻 ) ∣ ( ( iEdg ‘ 𝐻 ) ‘ 𝑥 ) = { 𝑢 } } ) ) = ( ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑢 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) = { 𝑢 } } ) ) ) |
14 |
3 13
|
mpteq12dv |
⊢ ( 𝜑 → ( 𝑢 ∈ ( Vtx ‘ 𝐻 ) ↦ ( ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐻 ) ∣ 𝑢 ∈ ( ( iEdg ‘ 𝐻 ) ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐻 ) ∣ ( ( iEdg ‘ 𝐻 ) ‘ 𝑥 ) = { 𝑢 } } ) ) ) = ( 𝑢 ∈ ( Vtx ‘ 𝐺 ) ↦ ( ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑢 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) = { 𝑢 } } ) ) ) ) |
15 |
|
eqid |
⊢ ( Vtx ‘ 𝐻 ) = ( Vtx ‘ 𝐻 ) |
16 |
|
eqid |
⊢ ( iEdg ‘ 𝐻 ) = ( iEdg ‘ 𝐻 ) |
17 |
|
eqid |
⊢ dom ( iEdg ‘ 𝐻 ) = dom ( iEdg ‘ 𝐻 ) |
18 |
15 16 17
|
vtxdgfval |
⊢ ( 𝐻 ∈ 𝑌 → ( VtxDeg ‘ 𝐻 ) = ( 𝑢 ∈ ( Vtx ‘ 𝐻 ) ↦ ( ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐻 ) ∣ 𝑢 ∈ ( ( iEdg ‘ 𝐻 ) ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐻 ) ∣ ( ( iEdg ‘ 𝐻 ) ‘ 𝑥 ) = { 𝑢 } } ) ) ) ) |
19 |
2 18
|
syl |
⊢ ( 𝜑 → ( VtxDeg ‘ 𝐻 ) = ( 𝑢 ∈ ( Vtx ‘ 𝐻 ) ↦ ( ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐻 ) ∣ 𝑢 ∈ ( ( iEdg ‘ 𝐻 ) ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐻 ) ∣ ( ( iEdg ‘ 𝐻 ) ‘ 𝑥 ) = { 𝑢 } } ) ) ) ) |
20 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
21 |
|
eqid |
⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) |
22 |
|
eqid |
⊢ dom ( iEdg ‘ 𝐺 ) = dom ( iEdg ‘ 𝐺 ) |
23 |
20 21 22
|
vtxdgfval |
⊢ ( 𝐺 ∈ 𝑋 → ( VtxDeg ‘ 𝐺 ) = ( 𝑢 ∈ ( Vtx ‘ 𝐺 ) ↦ ( ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑢 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) = { 𝑢 } } ) ) ) ) |
24 |
1 23
|
syl |
⊢ ( 𝜑 → ( VtxDeg ‘ 𝐺 ) = ( 𝑢 ∈ ( Vtx ‘ 𝐺 ) ↦ ( ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑢 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) = { 𝑢 } } ) ) ) ) |
25 |
14 19 24
|
3eqtr4d |
⊢ ( 𝜑 → ( VtxDeg ‘ 𝐻 ) = ( VtxDeg ‘ 𝐺 ) ) |