Step |
Hyp |
Ref |
Expression |
1 |
|
clnbgr3stgrgrlic.n |
⊢ 𝑁 ∈ ℕ0 |
2 |
|
clnbgr3stgrgrlic.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
3 |
|
clnbgr3stgrgrlic.w |
⊢ 𝑊 = ( Vtx ‘ 𝐻 ) |
4 |
2
|
fvexi |
⊢ 𝑉 ∈ V |
5 |
3
|
fvexi |
⊢ 𝑊 ∈ V |
6 |
4 5
|
pm3.2i |
⊢ ( 𝑉 ∈ V ∧ 𝑊 ∈ V ) |
7 |
|
breng |
⊢ ( ( 𝑉 ∈ V ∧ 𝑊 ∈ V ) → ( 𝑉 ≈ 𝑊 ↔ ∃ 𝑓 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ) |
8 |
6 7
|
mp1i |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) → ( 𝑉 ≈ 𝑊 ↔ ∃ 𝑓 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ) |
9 |
|
usgruhgr |
⊢ ( 𝐻 ∈ USGraph → 𝐻 ∈ UHGraph ) |
10 |
9
|
adantl |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) → 𝐻 ∈ UHGraph ) |
11 |
10
|
3ad2ant1 |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) → 𝐻 ∈ UHGraph ) |
12 |
3
|
clnbgrssvtx |
⊢ ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ⊆ 𝑊 |
13 |
12
|
a1i |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) ∧ 𝑥 ∈ 𝑉 ) → ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ⊆ 𝑊 ) |
14 |
3
|
isubgruhgr |
⊢ ( ( 𝐻 ∈ UHGraph ∧ ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ⊆ 𝑊 ) → ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ∈ UHGraph ) |
15 |
11 13 14
|
syl2an2r |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) ∧ 𝑥 ∈ 𝑉 ) → ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ∈ UHGraph ) |
16 |
|
f1of |
⊢ ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 → 𝑓 : 𝑉 ⟶ 𝑊 ) |
17 |
16
|
3ad2ant2 |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ 𝑥 ∈ 𝑉 ) → 𝑓 : 𝑉 ⟶ 𝑊 ) |
18 |
|
simp3 |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ 𝑥 ∈ 𝑉 ) → 𝑥 ∈ 𝑉 ) |
19 |
17 18
|
ffvelcdmd |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ 𝑥 ∈ 𝑉 ) → ( 𝑓 ‘ 𝑥 ) ∈ 𝑊 ) |
20 |
|
oveq2 |
⊢ ( 𝑦 = ( 𝑓 ‘ 𝑥 ) → ( 𝐻 ClNeighbVtx 𝑦 ) = ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) |
21 |
20
|
oveq2d |
⊢ ( 𝑦 = ( 𝑓 ‘ 𝑥 ) → ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) = ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) |
22 |
21
|
breq1d |
⊢ ( 𝑦 = ( 𝑓 ‘ 𝑥 ) → ( ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ↔ ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) ) |
23 |
22
|
rspcv |
⊢ ( ( 𝑓 ‘ 𝑥 ) ∈ 𝑊 → ( ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) → ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) ) |
24 |
19 23
|
syl |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ 𝑥 ∈ 𝑉 ) → ( ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) → ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) ) |
25 |
24
|
3exp |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) → ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 → ( 𝑥 ∈ 𝑉 → ( ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) → ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) ) ) ) |
26 |
25
|
com34 |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) → ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 → ( ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) → ( 𝑥 ∈ 𝑉 → ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) ) ) ) |
27 |
26
|
3imp1 |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) ∧ 𝑥 ∈ 𝑉 ) → ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) |
28 |
|
gricsym |
⊢ ( ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ∈ UHGraph → ( ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) → ( StarGr ‘ 𝑁 ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) |
29 |
15 27 28
|
sylc |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) ∧ 𝑥 ∈ 𝑉 ) → ( StarGr ‘ 𝑁 ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) |
30 |
29
|
anim1ci |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) ∧ 𝑥 ∈ 𝑉 ) ∧ ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) → ( ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ∧ ( StarGr ‘ 𝑁 ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) |
31 |
|
grictr |
⊢ ( ( ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ∧ ( StarGr ‘ 𝑁 ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) → ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) |
32 |
30 31
|
syl |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) ∧ 𝑥 ∈ 𝑉 ) ∧ ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) → ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) |
33 |
32
|
ex |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) ∧ 𝑥 ∈ 𝑉 ) → ( ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) → ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) |
34 |
33
|
ralimdva |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) → ( ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) → ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) |
35 |
34
|
3exp |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) → ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 → ( ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) → ( ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) → ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) ) ) |
36 |
35
|
com24 |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) → ( ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) → ( ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) → ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 → ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) ) ) |
37 |
36
|
imp32 |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ ( ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) ) → ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 → ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) |
38 |
37
|
ancld |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ ( ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) ) → ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 → ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) ) |
39 |
38
|
eximdv |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) ∧ ( ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) ) → ( ∃ 𝑓 𝑓 : 𝑉 –1-1-onto→ 𝑊 → ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) ) |
40 |
39
|
ex |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) → ( ( ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) → ( ∃ 𝑓 𝑓 : 𝑉 –1-1-onto→ 𝑊 → ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) ) ) |
41 |
40
|
com23 |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) → ( ∃ 𝑓 𝑓 : 𝑉 –1-1-onto→ 𝑊 → ( ( ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) → ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) ) ) |
42 |
8 41
|
sylbid |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) → ( 𝑉 ≈ 𝑊 → ( ( ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) → ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) ) ) |
43 |
42
|
3impia |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊 ) → ( ( ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) → ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) ) |
44 |
43
|
3impib |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊 ) ∧ ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) → ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) |
45 |
2 3
|
dfgrlic2 |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ) → ( 𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) ) |
46 |
45
|
3adant3 |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊 ) → ( 𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) ) |
47 |
46
|
3ad2ant1 |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊 ) ∧ ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) → ( 𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑥 ) ) ) ) ) ) |
48 |
44 47
|
mpbird |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊 ) ∧ ∀ 𝑥 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑥 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ∧ ∀ 𝑦 ∈ 𝑊 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx 𝑦 ) ) ≃𝑔𝑟 ( StarGr ‘ 𝑁 ) ) → 𝐺 ≃𝑙𝑔𝑟 𝐻 ) |