Step |
Hyp |
Ref |
Expression |
1 |
|
isubgr3stgr.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
isubgr3stgr.u |
⊢ 𝑈 = ( 𝐺 NeighbVtx 𝑋 ) |
3 |
|
isubgr3stgr.c |
⊢ 𝐶 = ( 𝐺 ClNeighbVtx 𝑋 ) |
4 |
|
isubgr3stgr.n |
⊢ 𝑁 ∈ ℕ0 |
5 |
|
isubgr3stgr.s |
⊢ 𝑆 = ( StarGr ‘ 𝑁 ) |
6 |
|
isubgr3stgr.w |
⊢ 𝑊 = ( Vtx ‘ 𝑆 ) |
7 |
|
isubgr3stgr.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
8 |
|
isubgr3stgr.i |
⊢ 𝐼 = ( Edg ‘ ( 𝐺 ISubGr 𝐶 ) ) |
9 |
|
isubgr3stgr.h |
⊢ 𝐻 = ( 𝑖 ∈ 𝐼 ↦ ( 𝐹 “ 𝑖 ) ) |
10 |
|
usgruhgr |
⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ UHGraph ) |
11 |
10
|
adantr |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) → 𝐺 ∈ UHGraph ) |
12 |
11
|
adantr |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) → 𝐺 ∈ UHGraph ) |
13 |
1
|
clnbgrssvtx |
⊢ ( 𝐺 ClNeighbVtx 𝑋 ) ⊆ 𝑉 |
14 |
3 13
|
eqsstri |
⊢ 𝐶 ⊆ 𝑉 |
15 |
14
|
a1i |
⊢ ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) → 𝐶 ⊆ 𝑉 ) |
16 |
|
eqid |
⊢ ( 𝐺 ISubGr 𝐶 ) = ( 𝐺 ISubGr 𝐶 ) |
17 |
1 7 16 8
|
isubgredg |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝐶 ⊆ 𝑉 ) → ( 𝑖 ∈ 𝐼 ↔ ( 𝑖 ∈ 𝐸 ∧ 𝑖 ⊆ 𝐶 ) ) ) |
18 |
12 15 17
|
syl2an |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ( 𝑖 ∈ 𝐼 ↔ ( 𝑖 ∈ 𝐸 ∧ 𝑖 ⊆ 𝐶 ) ) ) |
19 |
|
f1of |
⊢ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 → 𝐹 : 𝐶 ⟶ 𝑊 ) |
20 |
5
|
fveq2i |
⊢ ( Vtx ‘ 𝑆 ) = ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) |
21 |
|
stgrvtx |
⊢ ( 𝑁 ∈ ℕ0 → ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) = ( 0 ... 𝑁 ) ) |
22 |
4 21
|
ax-mp |
⊢ ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) = ( 0 ... 𝑁 ) |
23 |
6 20 22
|
3eqtri |
⊢ 𝑊 = ( 0 ... 𝑁 ) |
24 |
23
|
eqimssi |
⊢ 𝑊 ⊆ ( 0 ... 𝑁 ) |
25 |
24
|
a1i |
⊢ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 → 𝑊 ⊆ ( 0 ... 𝑁 ) ) |
26 |
19 25
|
fssd |
⊢ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 → 𝐹 : 𝐶 ⟶ ( 0 ... 𝑁 ) ) |
27 |
26
|
ad2antrl |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → 𝐹 : 𝐶 ⟶ ( 0 ... 𝑁 ) ) |
28 |
27
|
adantr |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑖 ∈ 𝐸 ∧ 𝑖 ⊆ 𝐶 ) ) → 𝐹 : 𝐶 ⟶ ( 0 ... 𝑁 ) ) |
29 |
28
|
fimassd |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑖 ∈ 𝐸 ∧ 𝑖 ⊆ 𝐶 ) ) → ( 𝐹 “ 𝑖 ) ⊆ ( 0 ... 𝑁 ) ) |
30 |
|
simplll |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → 𝐺 ∈ USGraph ) |
31 |
|
simpl |
⊢ ( ( 𝑖 ∈ 𝐸 ∧ 𝑖 ⊆ 𝐶 ) → 𝑖 ∈ 𝐸 ) |
32 |
1 7
|
usgredg |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑖 ∈ 𝐸 ) → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑖 = { 𝑎 , 𝑏 } ) ) |
33 |
30 31 32
|
syl2an |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑖 ∈ 𝐸 ∧ 𝑖 ⊆ 𝐶 ) ) → ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑖 = { 𝑎 , 𝑏 } ) ) |
34 |
|
vex |
⊢ 𝑎 ∈ V |
35 |
|
vex |
⊢ 𝑏 ∈ V |
36 |
34 35
|
prss |
⊢ ( ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ↔ { 𝑎 , 𝑏 } ⊆ 𝐶 ) |
37 |
|
elclnbgrelnbgr |
⊢ ( ( 𝑎 ∈ ( 𝐺 ClNeighbVtx 𝑋 ) ∧ 𝑎 ≠ 𝑋 ) → 𝑎 ∈ ( 𝐺 NeighbVtx 𝑋 ) ) |
38 |
37
|
expcom |
⊢ ( 𝑎 ≠ 𝑋 → ( 𝑎 ∈ ( 𝐺 ClNeighbVtx 𝑋 ) → 𝑎 ∈ ( 𝐺 NeighbVtx 𝑋 ) ) ) |
39 |
3
|
eleq2i |
⊢ ( 𝑎 ∈ 𝐶 ↔ 𝑎 ∈ ( 𝐺 ClNeighbVtx 𝑋 ) ) |
40 |
2
|
eleq2i |
⊢ ( 𝑎 ∈ 𝑈 ↔ 𝑎 ∈ ( 𝐺 NeighbVtx 𝑋 ) ) |
41 |
38 39 40
|
3imtr4g |
⊢ ( 𝑎 ≠ 𝑋 → ( 𝑎 ∈ 𝐶 → 𝑎 ∈ 𝑈 ) ) |
42 |
|
elclnbgrelnbgr |
⊢ ( ( 𝑏 ∈ ( 𝐺 ClNeighbVtx 𝑋 ) ∧ 𝑏 ≠ 𝑋 ) → 𝑏 ∈ ( 𝐺 NeighbVtx 𝑋 ) ) |
43 |
42
|
expcom |
⊢ ( 𝑏 ≠ 𝑋 → ( 𝑏 ∈ ( 𝐺 ClNeighbVtx 𝑋 ) → 𝑏 ∈ ( 𝐺 NeighbVtx 𝑋 ) ) ) |
44 |
3
|
eleq2i |
⊢ ( 𝑏 ∈ 𝐶 ↔ 𝑏 ∈ ( 𝐺 ClNeighbVtx 𝑋 ) ) |
45 |
2
|
eleq2i |
⊢ ( 𝑏 ∈ 𝑈 ↔ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑋 ) ) |
46 |
43 44 45
|
3imtr4g |
⊢ ( 𝑏 ≠ 𝑋 → ( 𝑏 ∈ 𝐶 → 𝑏 ∈ 𝑈 ) ) |
47 |
41 46
|
im2anan9r |
⊢ ( ( 𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋 ) → ( ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) → ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ) ) |
48 |
47
|
imp |
⊢ ( ( ( 𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋 ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ) |
49 |
48
|
3adant3 |
⊢ ( ( ( 𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋 ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) → ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) ) |
50 |
|
preq1 |
⊢ ( 𝑥 = 𝑎 → { 𝑥 , 𝑦 } = { 𝑎 , 𝑦 } ) |
51 |
|
eqidd |
⊢ ( 𝑥 = 𝑎 → 𝐸 = 𝐸 ) |
52 |
50 51
|
neleq12d |
⊢ ( 𝑥 = 𝑎 → ( { 𝑥 , 𝑦 } ∉ 𝐸 ↔ { 𝑎 , 𝑦 } ∉ 𝐸 ) ) |
53 |
|
preq2 |
⊢ ( 𝑦 = 𝑏 → { 𝑎 , 𝑦 } = { 𝑎 , 𝑏 } ) |
54 |
|
eqidd |
⊢ ( 𝑦 = 𝑏 → 𝐸 = 𝐸 ) |
55 |
53 54
|
neleq12d |
⊢ ( 𝑦 = 𝑏 → ( { 𝑎 , 𝑦 } ∉ 𝐸 ↔ { 𝑎 , 𝑏 } ∉ 𝐸 ) ) |
56 |
52 55
|
rspc2v |
⊢ ( ( 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈 ) → ( ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 → { 𝑎 , 𝑏 } ∉ 𝐸 ) ) |
57 |
49 56
|
syl |
⊢ ( ( ( 𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋 ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) → ( ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 → { 𝑎 , 𝑏 } ∉ 𝐸 ) ) |
58 |
|
pm2.24nel |
⊢ ( { 𝑎 , 𝑏 } ∈ 𝐸 → ( { 𝑎 , 𝑏 } ∉ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) |
59 |
58
|
adantl |
⊢ ( ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ( { 𝑎 , 𝑏 } ∉ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) |
60 |
59
|
3ad2ant3 |
⊢ ( ( ( 𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋 ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) → ( { 𝑎 , 𝑏 } ∉ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) |
61 |
57 60
|
syld |
⊢ ( ( ( 𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋 ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) → ( ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) |
62 |
61
|
3exp |
⊢ ( ( 𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋 ) → ( ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) → ( ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ( ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) ) |
63 |
62
|
com24 |
⊢ ( ( 𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋 ) → ( ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 → ( ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ( ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) ) |
64 |
63
|
adantld |
⊢ ( ( 𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋 ) → ( ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) → ( ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ( ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) ) |
65 |
64
|
adantld |
⊢ ( ( 𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋 ) → ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) → ( ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ( ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) ) |
66 |
65
|
adantrd |
⊢ ( ( 𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋 ) → ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ( ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ( ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) ) |
67 |
66
|
imp4c |
⊢ ( ( 𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋 ) → ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) |
68 |
|
simpl |
⊢ ( ( 𝑏 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) ) → 𝑏 = 𝑋 ) |
69 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) |
70 |
69
|
adantl |
⊢ ( ( 𝑏 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) ) → ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) |
71 |
|
simplrl |
⊢ ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → 𝑎 ≠ 𝑏 ) |
72 |
71
|
necomd |
⊢ ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → 𝑏 ≠ 𝑎 ) |
73 |
72
|
adantl |
⊢ ( ( 𝑏 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) ) → 𝑏 ≠ 𝑎 ) |
74 |
|
simprrr |
⊢ ( ( 𝑏 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) ) → 𝑏 ∈ 𝐶 ) |
75 |
|
simprrl |
⊢ ( ( 𝑏 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) ) → 𝑎 ∈ 𝐶 ) |
76 |
1 2 3 4 5 6 7
|
isubgr3stgrlem4 |
⊢ ( ( 𝑏 = 𝑋 ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑏 ≠ 𝑎 ∧ 𝑏 ∈ 𝐶 ∧ 𝑎 ∈ 𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑏 , 𝑎 } ) = { 0 , 𝑧 } ) |
77 |
68 70 73 74 75 76
|
syl113anc |
⊢ ( ( 𝑏 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑏 , 𝑎 } ) = { 0 , 𝑧 } ) |
78 |
|
prcom |
⊢ { 𝑎 , 𝑏 } = { 𝑏 , 𝑎 } |
79 |
78
|
imaeq2i |
⊢ ( 𝐹 “ { 𝑎 , 𝑏 } ) = ( 𝐹 “ { 𝑏 , 𝑎 } ) |
80 |
79
|
eqeq1i |
⊢ ( ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ↔ ( 𝐹 “ { 𝑏 , 𝑎 } ) = { 0 , 𝑧 } ) |
81 |
80
|
rexbii |
⊢ ( ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ↔ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑏 , 𝑎 } ) = { 0 , 𝑧 } ) |
82 |
77 81
|
sylibr |
⊢ ( ( 𝑏 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) |
83 |
82
|
ex |
⊢ ( 𝑏 = 𝑋 → ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) |
84 |
|
simpl |
⊢ ( ( 𝑎 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) ) → 𝑎 = 𝑋 ) |
85 |
69
|
adantl |
⊢ ( ( 𝑎 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) ) → ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) |
86 |
71
|
adantl |
⊢ ( ( 𝑎 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) ) → 𝑎 ≠ 𝑏 ) |
87 |
|
simprrl |
⊢ ( ( 𝑎 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) ) → 𝑎 ∈ 𝐶 ) |
88 |
|
simprrr |
⊢ ( ( 𝑎 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) ) → 𝑏 ∈ 𝐶 ) |
89 |
1 2 3 4 5 6 7
|
isubgr3stgrlem4 |
⊢ ( ( 𝑎 = 𝑋 ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) |
90 |
84 85 86 87 88 89
|
syl113anc |
⊢ ( ( 𝑎 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) |
91 |
90
|
ex |
⊢ ( 𝑎 = 𝑋 → ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) |
92 |
67 83 91
|
pm2.61iine |
⊢ ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) |
93 |
92
|
ex |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) → ( ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) |
94 |
36 93
|
biimtrrid |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) → ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) |
95 |
94
|
exp32 |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ( 𝑎 ≠ 𝑏 → ( { 𝑎 , 𝑏 } ∈ 𝐸 → ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) ) |
96 |
95
|
adantrd |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ( ( 𝑎 ≠ 𝑏 ∧ 𝑖 = { 𝑎 , 𝑏 } ) → ( { 𝑎 , 𝑏 } ∈ 𝐸 → ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) ) |
97 |
96
|
imp |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑖 = { 𝑎 , 𝑏 } ) ) → ( { 𝑎 , 𝑏 } ∈ 𝐸 → ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) |
98 |
97
|
com23 |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑖 = { 𝑎 , 𝑏 } ) ) → ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ( { 𝑎 , 𝑏 } ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) |
99 |
|
sseq1 |
⊢ ( 𝑖 = { 𝑎 , 𝑏 } → ( 𝑖 ⊆ 𝐶 ↔ { 𝑎 , 𝑏 } ⊆ 𝐶 ) ) |
100 |
|
eleq1 |
⊢ ( 𝑖 = { 𝑎 , 𝑏 } → ( 𝑖 ∈ 𝐸 ↔ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
101 |
|
imaeq2 |
⊢ ( 𝑖 = { 𝑎 , 𝑏 } → ( 𝐹 “ 𝑖 ) = ( 𝐹 “ { 𝑎 , 𝑏 } ) ) |
102 |
101
|
eqeq1d |
⊢ ( 𝑖 = { 𝑎 , 𝑏 } → ( ( 𝐹 “ 𝑖 ) = { 0 , 𝑧 } ↔ ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) |
103 |
102
|
rexbidv |
⊢ ( 𝑖 = { 𝑎 , 𝑏 } → ( ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ 𝑖 ) = { 0 , 𝑧 } ↔ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) |
104 |
100 103
|
imbi12d |
⊢ ( 𝑖 = { 𝑎 , 𝑏 } → ( ( 𝑖 ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ 𝑖 ) = { 0 , 𝑧 } ) ↔ ( { 𝑎 , 𝑏 } ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) |
105 |
99 104
|
imbi12d |
⊢ ( 𝑖 = { 𝑎 , 𝑏 } → ( ( 𝑖 ⊆ 𝐶 → ( 𝑖 ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ 𝑖 ) = { 0 , 𝑧 } ) ) ↔ ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ( { 𝑎 , 𝑏 } ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) ) |
106 |
105
|
adantl |
⊢ ( ( 𝑎 ≠ 𝑏 ∧ 𝑖 = { 𝑎 , 𝑏 } ) → ( ( 𝑖 ⊆ 𝐶 → ( 𝑖 ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ 𝑖 ) = { 0 , 𝑧 } ) ) ↔ ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ( { 𝑎 , 𝑏 } ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) ) |
107 |
106
|
adantl |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑖 = { 𝑎 , 𝑏 } ) ) → ( ( 𝑖 ⊆ 𝐶 → ( 𝑖 ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ 𝑖 ) = { 0 , 𝑧 } ) ) ↔ ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ( { 𝑎 , 𝑏 } ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) ) |
108 |
98 107
|
mpbird |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑖 = { 𝑎 , 𝑏 } ) ) → ( 𝑖 ⊆ 𝐶 → ( 𝑖 ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ 𝑖 ) = { 0 , 𝑧 } ) ) ) |
109 |
108
|
ex |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ( ( 𝑎 ≠ 𝑏 ∧ 𝑖 = { 𝑎 , 𝑏 } ) → ( 𝑖 ⊆ 𝐶 → ( 𝑖 ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ 𝑖 ) = { 0 , 𝑧 } ) ) ) ) |
110 |
109
|
com24 |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ( 𝑖 ∈ 𝐸 → ( 𝑖 ⊆ 𝐶 → ( ( 𝑎 ≠ 𝑏 ∧ 𝑖 = { 𝑎 , 𝑏 } ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ 𝑖 ) = { 0 , 𝑧 } ) ) ) ) |
111 |
110
|
imp32 |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑖 ∈ 𝐸 ∧ 𝑖 ⊆ 𝐶 ) ) → ( ( 𝑎 ≠ 𝑏 ∧ 𝑖 = { 𝑎 , 𝑏 } ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ 𝑖 ) = { 0 , 𝑧 } ) ) |
112 |
111
|
a1d |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑖 ∈ 𝐸 ∧ 𝑖 ⊆ 𝐶 ) ) → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → ( ( 𝑎 ≠ 𝑏 ∧ 𝑖 = { 𝑎 , 𝑏 } ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ 𝑖 ) = { 0 , 𝑧 } ) ) ) |
113 |
112
|
rexlimdvv |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑖 ∈ 𝐸 ∧ 𝑖 ⊆ 𝐶 ) ) → ( ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑖 = { 𝑎 , 𝑏 } ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ 𝑖 ) = { 0 , 𝑧 } ) ) |
114 |
33 113
|
mpd |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑖 ∈ 𝐸 ∧ 𝑖 ⊆ 𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ 𝑖 ) = { 0 , 𝑧 } ) |
115 |
|
stgredgel |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝐹 “ 𝑖 ) ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) ↔ ( ( 𝐹 “ 𝑖 ) ⊆ ( 0 ... 𝑁 ) ∧ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ 𝑖 ) = { 0 , 𝑧 } ) ) ) |
116 |
4 115
|
ax-mp |
⊢ ( ( 𝐹 “ 𝑖 ) ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) ↔ ( ( 𝐹 “ 𝑖 ) ⊆ ( 0 ... 𝑁 ) ∧ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ 𝑖 ) = { 0 , 𝑧 } ) ) |
117 |
29 114 116
|
sylanbrc |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ ( 𝑖 ∈ 𝐸 ∧ 𝑖 ⊆ 𝐶 ) ) → ( 𝐹 “ 𝑖 ) ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) ) |
118 |
18 117
|
sylbida |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑖 ∈ 𝐼 ) → ( 𝐹 “ 𝑖 ) ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) ) |
119 |
118 9
|
fmptd |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥 ∈ 𝑈 ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → 𝐻 : 𝐼 ⟶ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) ) |