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 |
|
stgredgel |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝐽 ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) ↔ ( 𝐽 ⊆ ( 0 ... 𝑁 ) ∧ ∃ 𝑦 ∈ ( 1 ... 𝑁 ) 𝐽 = { 0 , 𝑦 } ) ) ) |
11 |
4 10
|
mp1i |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ( 𝐽 ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) ↔ ( 𝐽 ⊆ ( 0 ... 𝑁 ) ∧ ∃ 𝑦 ∈ ( 1 ... 𝑁 ) 𝐽 = { 0 , 𝑦 } ) ) ) |
12 |
|
c0ex |
⊢ 0 ∈ V |
13 |
12
|
a1i |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → 0 ∈ V ) |
14 |
|
prssg |
⊢ ( ( 0 ∈ V ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( 0 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 0 ... 𝑁 ) ) ↔ { 0 , 𝑦 } ⊆ ( 0 ... 𝑁 ) ) ) |
15 |
13 14
|
sylan |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( 0 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 0 ... 𝑁 ) ) ↔ { 0 , 𝑦 } ⊆ ( 0 ... 𝑁 ) ) ) |
16 |
|
f1ocnv |
⊢ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 → ◡ 𝐹 : 𝑊 –1-1-onto→ 𝐶 ) |
17 |
|
f1ofn |
⊢ ( ◡ 𝐹 : 𝑊 –1-1-onto→ 𝐶 → ◡ 𝐹 Fn 𝑊 ) |
18 |
5
|
fveq2i |
⊢ ( Vtx ‘ 𝑆 ) = ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) |
19 |
|
stgrvtx |
⊢ ( 𝑁 ∈ ℕ0 → ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) = ( 0 ... 𝑁 ) ) |
20 |
4 19
|
ax-mp |
⊢ ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) = ( 0 ... 𝑁 ) |
21 |
6 18 20
|
3eqtri |
⊢ 𝑊 = ( 0 ... 𝑁 ) |
22 |
21
|
fneq2i |
⊢ ( ◡ 𝐹 Fn 𝑊 ↔ ◡ 𝐹 Fn ( 0 ... 𝑁 ) ) |
23 |
17 22
|
sylib |
⊢ ( ◡ 𝐹 : 𝑊 –1-1-onto→ 𝐶 → ◡ 𝐹 Fn ( 0 ... 𝑁 ) ) |
24 |
16 23
|
syl |
⊢ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 → ◡ 𝐹 Fn ( 0 ... 𝑁 ) ) |
25 |
24
|
ad2antrl |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ◡ 𝐹 Fn ( 0 ... 𝑁 ) ) |
26 |
25
|
adantr |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ◡ 𝐹 Fn ( 0 ... 𝑁 ) ) |
27 |
26
|
anim1i |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( 0 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 0 ... 𝑁 ) ) ) → ( ◡ 𝐹 Fn ( 0 ... 𝑁 ) ∧ ( 0 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 0 ... 𝑁 ) ) ) ) |
28 |
|
3anass |
⊢ ( ( ◡ 𝐹 Fn ( 0 ... 𝑁 ) ∧ 0 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 0 ... 𝑁 ) ) ↔ ( ◡ 𝐹 Fn ( 0 ... 𝑁 ) ∧ ( 0 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 0 ... 𝑁 ) ) ) ) |
29 |
27 28
|
sylibr |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( 0 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 0 ... 𝑁 ) ) ) → ( ◡ 𝐹 Fn ( 0 ... 𝑁 ) ∧ 0 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 0 ... 𝑁 ) ) ) |
30 |
29
|
ex |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( 0 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 0 ... 𝑁 ) ) → ( ◡ 𝐹 Fn ( 0 ... 𝑁 ) ∧ 0 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 0 ... 𝑁 ) ) ) ) |
31 |
15 30
|
sylbird |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( { 0 , 𝑦 } ⊆ ( 0 ... 𝑁 ) → ( ◡ 𝐹 Fn ( 0 ... 𝑁 ) ∧ 0 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 0 ... 𝑁 ) ) ) ) |
32 |
31
|
imp |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ { 0 , 𝑦 } ⊆ ( 0 ... 𝑁 ) ) → ( ◡ 𝐹 Fn ( 0 ... 𝑁 ) ∧ 0 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 0 ... 𝑁 ) ) ) |
33 |
|
fnimapr |
⊢ ( ( ◡ 𝐹 Fn ( 0 ... 𝑁 ) ∧ 0 ∈ ( 0 ... 𝑁 ) ∧ 𝑦 ∈ ( 0 ... 𝑁 ) ) → ( ◡ 𝐹 “ { 0 , 𝑦 } ) = { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ) |
34 |
32 33
|
syl |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ { 0 , 𝑦 } ⊆ ( 0 ... 𝑁 ) ) → ( ◡ 𝐹 “ { 0 , 𝑦 } ) = { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ) |
35 |
1
|
clnbgrvtxel |
⊢ ( 𝑋 ∈ 𝑉 → 𝑋 ∈ ( 𝐺 ClNeighbVtx 𝑋 ) ) |
36 |
35
|
adantl |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ ( 𝐺 ClNeighbVtx 𝑋 ) ) |
37 |
36 3
|
eleqtrrdi |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ 𝐶 ) |
38 |
|
simpl |
⊢ ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) → 𝐹 : 𝐶 –1-1-onto→ 𝑊 ) |
39 |
37 38
|
anim12ci |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ 𝑋 ∈ 𝐶 ) ) |
40 |
|
simprr |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ( 𝐹 ‘ 𝑋 ) = 0 ) |
41 |
39 40
|
jca |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) |
42 |
41
|
adantr |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) |
43 |
|
f1ocnvfv |
⊢ ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ 𝑋 ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝑋 ) = 0 → ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) ) |
44 |
43
|
imp |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ 𝑋 ∈ 𝐶 ) ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) → ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) |
45 |
42 44
|
syl |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) |
46 |
35 3
|
eleqtrrdi |
⊢ ( 𝑋 ∈ 𝑉 → 𝑋 ∈ 𝐶 ) |
47 |
46
|
ad3antlr |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → 𝑋 ∈ 𝐶 ) |
48 |
|
f1of |
⊢ ( ◡ 𝐹 : 𝑊 –1-1-onto→ 𝐶 → ◡ 𝐹 : 𝑊 ⟶ 𝐶 ) |
49 |
16 48
|
syl |
⊢ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 → ◡ 𝐹 : 𝑊 ⟶ 𝐶 ) |
50 |
49
|
ad2antrl |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ◡ 𝐹 : 𝑊 ⟶ 𝐶 ) |
51 |
50
|
adantr |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ◡ 𝐹 : 𝑊 ⟶ 𝐶 ) |
52 |
|
fz1ssfz0 |
⊢ ( 1 ... 𝑁 ) ⊆ ( 0 ... 𝑁 ) |
53 |
52
|
sseli |
⊢ ( 𝑦 ∈ ( 1 ... 𝑁 ) → 𝑦 ∈ ( 0 ... 𝑁 ) ) |
54 |
53 21
|
eleqtrrdi |
⊢ ( 𝑦 ∈ ( 1 ... 𝑁 ) → 𝑦 ∈ 𝑊 ) |
55 |
54
|
adantl |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → 𝑦 ∈ 𝑊 ) |
56 |
51 55
|
ffvelcdmd |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ◡ 𝐹 ‘ 𝑦 ) ∈ 𝐶 ) |
57 |
47 56
|
jca |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( 𝑋 ∈ 𝐶 ∧ ( ◡ 𝐹 ‘ 𝑦 ) ∈ 𝐶 ) ) |
58 |
3
|
eleq2i |
⊢ ( ( ◡ 𝐹 ‘ 𝑦 ) ∈ 𝐶 ↔ ( ◡ 𝐹 ‘ 𝑦 ) ∈ ( 𝐺 ClNeighbVtx 𝑋 ) ) |
59 |
|
usgrupgr |
⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ UPGraph ) |
60 |
59
|
anim1i |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) → ( 𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉 ) ) |
61 |
60
|
ad2antrr |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( 𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉 ) ) |
62 |
1
|
clnbgrssvtx |
⊢ ( 𝐺 ClNeighbVtx 𝑋 ) ⊆ 𝑉 |
63 |
3 62
|
eqsstri |
⊢ 𝐶 ⊆ 𝑉 |
64 |
63 56
|
sselid |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ◡ 𝐹 ‘ 𝑦 ) ∈ 𝑉 ) |
65 |
|
df-3an |
⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ◡ 𝐹 ‘ 𝑦 ) ∈ 𝑉 ) ↔ ( ( 𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( ◡ 𝐹 ‘ 𝑦 ) ∈ 𝑉 ) ) |
66 |
61 64 65
|
sylanbrc |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( 𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ◡ 𝐹 ‘ 𝑦 ) ∈ 𝑉 ) ) |
67 |
66
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) ∧ 𝑋 ∈ 𝐶 ) → ( 𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ◡ 𝐹 ‘ 𝑦 ) ∈ 𝑉 ) ) |
68 |
1 7
|
clnbupgrel |
⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ◡ 𝐹 ‘ 𝑦 ) ∈ 𝑉 ) → ( ( ◡ 𝐹 ‘ 𝑦 ) ∈ ( 𝐺 ClNeighbVtx 𝑋 ) ↔ ( ( ◡ 𝐹 ‘ 𝑦 ) = 𝑋 ∨ { ( ◡ 𝐹 ‘ 𝑦 ) , 𝑋 } ∈ 𝐸 ) ) ) |
69 |
67 68
|
syl |
⊢ ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) ∧ 𝑋 ∈ 𝐶 ) → ( ( ◡ 𝐹 ‘ 𝑦 ) ∈ ( 𝐺 ClNeighbVtx 𝑋 ) ↔ ( ( ◡ 𝐹 ‘ 𝑦 ) = 𝑋 ∨ { ( ◡ 𝐹 ‘ 𝑦 ) , 𝑋 } ∈ 𝐸 ) ) ) |
70 |
58 69
|
bitrid |
⊢ ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) ∧ 𝑋 ∈ 𝐶 ) → ( ( ◡ 𝐹 ‘ 𝑦 ) ∈ 𝐶 ↔ ( ( ◡ 𝐹 ‘ 𝑦 ) = 𝑋 ∨ { ( ◡ 𝐹 ‘ 𝑦 ) , 𝑋 } ∈ 𝐸 ) ) ) |
71 |
|
eqeq2 |
⊢ ( ( ◡ 𝐹 ‘ 0 ) = 𝑋 → ( ( ◡ 𝐹 ‘ 𝑦 ) = ( ◡ 𝐹 ‘ 0 ) ↔ ( ◡ 𝐹 ‘ 𝑦 ) = 𝑋 ) ) |
72 |
71
|
adantl |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) → ( ( ◡ 𝐹 ‘ 𝑦 ) = ( ◡ 𝐹 ‘ 0 ) ↔ ( ◡ 𝐹 ‘ 𝑦 ) = 𝑋 ) ) |
73 |
|
f1of1 |
⊢ ( ◡ 𝐹 : 𝑊 –1-1-onto→ 𝐶 → ◡ 𝐹 : 𝑊 –1-1→ 𝐶 ) |
74 |
16 73
|
syl |
⊢ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 → ◡ 𝐹 : 𝑊 –1-1→ 𝐶 ) |
75 |
74
|
ad2antrl |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ◡ 𝐹 : 𝑊 –1-1→ 𝐶 ) |
76 |
|
0elfz |
⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ ( 0 ... 𝑁 ) ) |
77 |
4 76
|
ax-mp |
⊢ 0 ∈ ( 0 ... 𝑁 ) |
78 |
77 21
|
eleqtrri |
⊢ 0 ∈ 𝑊 |
79 |
54 78
|
jctir |
⊢ ( 𝑦 ∈ ( 1 ... 𝑁 ) → ( 𝑦 ∈ 𝑊 ∧ 0 ∈ 𝑊 ) ) |
80 |
|
f1veqaeq |
⊢ ( ( ◡ 𝐹 : 𝑊 –1-1→ 𝐶 ∧ ( 𝑦 ∈ 𝑊 ∧ 0 ∈ 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑦 ) = ( ◡ 𝐹 ‘ 0 ) → 𝑦 = 0 ) ) |
81 |
75 79 80
|
syl2an |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( ◡ 𝐹 ‘ 𝑦 ) = ( ◡ 𝐹 ‘ 0 ) → 𝑦 = 0 ) ) |
82 |
|
elfznn |
⊢ ( 𝑦 ∈ ( 1 ... 𝑁 ) → 𝑦 ∈ ℕ ) |
83 |
|
nnne0 |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ≠ 0 ) |
84 |
|
eqneqall |
⊢ ( 𝑦 = 0 → ( 𝑦 ≠ 0 → { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ) ) |
85 |
83 84
|
syl5com |
⊢ ( 𝑦 ∈ ℕ → ( 𝑦 = 0 → { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ) ) |
86 |
82 85
|
syl |
⊢ ( 𝑦 ∈ ( 1 ... 𝑁 ) → ( 𝑦 = 0 → { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ) ) |
87 |
86
|
adantl |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( 𝑦 = 0 → { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ) ) |
88 |
81 87
|
syld |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( ( ◡ 𝐹 ‘ 𝑦 ) = ( ◡ 𝐹 ‘ 0 ) → { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ) ) |
89 |
88
|
adantr |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) → ( ( ◡ 𝐹 ‘ 𝑦 ) = ( ◡ 𝐹 ‘ 0 ) → { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ) ) |
90 |
72 89
|
sylbird |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) → ( ( ◡ 𝐹 ‘ 𝑦 ) = 𝑋 → { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ) ) |
91 |
90
|
adantr |
⊢ ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) ∧ 𝑋 ∈ 𝐶 ) → ( ( ◡ 𝐹 ‘ 𝑦 ) = 𝑋 → { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ) ) |
92 |
|
prcom |
⊢ { ( ◡ 𝐹 ‘ 𝑦 ) , 𝑋 } = { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } |
93 |
92
|
eleq1i |
⊢ ( { ( ◡ 𝐹 ‘ 𝑦 ) , 𝑋 } ∈ 𝐸 ↔ { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ) |
94 |
93
|
biimpi |
⊢ ( { ( ◡ 𝐹 ‘ 𝑦 ) , 𝑋 } ∈ 𝐸 → { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ) |
95 |
94
|
a1i |
⊢ ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) ∧ 𝑋 ∈ 𝐶 ) → ( { ( ◡ 𝐹 ‘ 𝑦 ) , 𝑋 } ∈ 𝐸 → { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ) ) |
96 |
91 95
|
jaod |
⊢ ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) ∧ 𝑋 ∈ 𝐶 ) → ( ( ( ◡ 𝐹 ‘ 𝑦 ) = 𝑋 ∨ { ( ◡ 𝐹 ‘ 𝑦 ) , 𝑋 } ∈ 𝐸 ) → { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ) ) |
97 |
70 96
|
sylbid |
⊢ ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) ∧ 𝑋 ∈ 𝐶 ) → ( ( ◡ 𝐹 ‘ 𝑦 ) ∈ 𝐶 → { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ) ) |
98 |
97
|
impr |
⊢ ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) ∧ ( 𝑋 ∈ 𝐶 ∧ ( ◡ 𝐹 ‘ 𝑦 ) ∈ 𝐶 ) ) → { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ) |
99 |
|
prssi |
⊢ ( ( 𝑋 ∈ 𝐶 ∧ ( ◡ 𝐹 ‘ 𝑦 ) ∈ 𝐶 ) → { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ⊆ 𝐶 ) |
100 |
99
|
adantl |
⊢ ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) ∧ ( 𝑋 ∈ 𝐶 ∧ ( ◡ 𝐹 ‘ 𝑦 ) ∈ 𝐶 ) ) → { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ⊆ 𝐶 ) |
101 |
98 100
|
jca |
⊢ ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) ∧ ( 𝑋 ∈ 𝐶 ∧ ( ◡ 𝐹 ‘ 𝑦 ) ∈ 𝐶 ) ) → ( { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ∧ { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ⊆ 𝐶 ) ) |
102 |
57 101
|
mpidan |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) → ( { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ∧ { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ⊆ 𝐶 ) ) |
103 |
|
preq1 |
⊢ ( ( ◡ 𝐹 ‘ 0 ) = 𝑋 → { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } = { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ) |
104 |
103
|
eleq1d |
⊢ ( ( ◡ 𝐹 ‘ 0 ) = 𝑋 → ( { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ↔ { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ) ) |
105 |
103
|
sseq1d |
⊢ ( ( ◡ 𝐹 ‘ 0 ) = 𝑋 → ( { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ⊆ 𝐶 ↔ { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ⊆ 𝐶 ) ) |
106 |
104 105
|
anbi12d |
⊢ ( ( ◡ 𝐹 ‘ 0 ) = 𝑋 → ( ( { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ∧ { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ⊆ 𝐶 ) ↔ ( { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ∧ { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ⊆ 𝐶 ) ) ) |
107 |
106
|
adantl |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) → ( ( { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ∧ { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ⊆ 𝐶 ) ↔ ( { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ∧ { 𝑋 , ( ◡ 𝐹 ‘ 𝑦 ) } ⊆ 𝐶 ) ) ) |
108 |
102 107
|
mpbird |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 0 ) = 𝑋 ) → ( { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ∧ { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ⊆ 𝐶 ) ) |
109 |
45 108
|
mpdan |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ∧ { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ⊆ 𝐶 ) ) |
110 |
109
|
adantr |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ { 0 , 𝑦 } ⊆ ( 0 ... 𝑁 ) ) → ( { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ∧ { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ⊆ 𝐶 ) ) |
111 |
|
usgruhgr |
⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ UHGraph ) |
112 |
111
|
ad3antrrr |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → 𝐺 ∈ UHGraph ) |
113 |
63
|
a1i |
⊢ ( { 0 , 𝑦 } ⊆ ( 0 ... 𝑁 ) → 𝐶 ⊆ 𝑉 ) |
114 |
|
eqid |
⊢ ( 𝐺 ISubGr 𝐶 ) = ( 𝐺 ISubGr 𝐶 ) |
115 |
1 7 114 8
|
isubgredg |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝐶 ⊆ 𝑉 ) → ( { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐼 ↔ ( { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ∧ { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ⊆ 𝐶 ) ) ) |
116 |
112 113 115
|
syl2an |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ { 0 , 𝑦 } ⊆ ( 0 ... 𝑁 ) ) → ( { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐼 ↔ ( { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐸 ∧ { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ⊆ 𝐶 ) ) ) |
117 |
110 116
|
mpbird |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ { 0 , 𝑦 } ⊆ ( 0 ... 𝑁 ) ) → { ( ◡ 𝐹 ‘ 0 ) , ( ◡ 𝐹 ‘ 𝑦 ) } ∈ 𝐼 ) |
118 |
34 117
|
eqeltrd |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) ∧ { 0 , 𝑦 } ⊆ ( 0 ... 𝑁 ) ) → ( ◡ 𝐹 “ { 0 , 𝑦 } ) ∈ 𝐼 ) |
119 |
118
|
ex |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( { 0 , 𝑦 } ⊆ ( 0 ... 𝑁 ) → ( ◡ 𝐹 “ { 0 , 𝑦 } ) ∈ 𝐼 ) ) |
120 |
|
sseq1 |
⊢ ( 𝐽 = { 0 , 𝑦 } → ( 𝐽 ⊆ ( 0 ... 𝑁 ) ↔ { 0 , 𝑦 } ⊆ ( 0 ... 𝑁 ) ) ) |
121 |
|
imaeq2 |
⊢ ( 𝐽 = { 0 , 𝑦 } → ( ◡ 𝐹 “ 𝐽 ) = ( ◡ 𝐹 “ { 0 , 𝑦 } ) ) |
122 |
121
|
eleq1d |
⊢ ( 𝐽 = { 0 , 𝑦 } → ( ( ◡ 𝐹 “ 𝐽 ) ∈ 𝐼 ↔ ( ◡ 𝐹 “ { 0 , 𝑦 } ) ∈ 𝐼 ) ) |
123 |
120 122
|
imbi12d |
⊢ ( 𝐽 = { 0 , 𝑦 } → ( ( 𝐽 ⊆ ( 0 ... 𝑁 ) → ( ◡ 𝐹 “ 𝐽 ) ∈ 𝐼 ) ↔ ( { 0 , 𝑦 } ⊆ ( 0 ... 𝑁 ) → ( ◡ 𝐹 “ { 0 , 𝑦 } ) ∈ 𝐼 ) ) ) |
124 |
119 123
|
syl5ibrcom |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) ∧ 𝑦 ∈ ( 1 ... 𝑁 ) ) → ( 𝐽 = { 0 , 𝑦 } → ( 𝐽 ⊆ ( 0 ... 𝑁 ) → ( ◡ 𝐹 “ 𝐽 ) ∈ 𝐼 ) ) ) |
125 |
124
|
rexlimdva |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ( ∃ 𝑦 ∈ ( 1 ... 𝑁 ) 𝐽 = { 0 , 𝑦 } → ( 𝐽 ⊆ ( 0 ... 𝑁 ) → ( ◡ 𝐹 “ 𝐽 ) ∈ 𝐼 ) ) ) |
126 |
125
|
impcomd |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ( ( 𝐽 ⊆ ( 0 ... 𝑁 ) ∧ ∃ 𝑦 ∈ ( 1 ... 𝑁 ) 𝐽 = { 0 , 𝑦 } ) → ( ◡ 𝐹 “ 𝐽 ) ∈ 𝐼 ) ) |
127 |
11 126
|
sylbid |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ) → ( 𝐽 ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) → ( ◡ 𝐹 “ 𝐽 ) ∈ 𝐼 ) ) |
128 |
127
|
3impia |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ 𝐽 ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) ) → ( ◡ 𝐹 “ 𝐽 ) ∈ 𝐼 ) |