| 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 ) ) ) |