Metamath Proof Explorer


Theorem isubgr3stgrlem6

Description: Lemma 6 for isubgr3stgr . (Contributed by AV, 24-Sep-2025)

Ref Expression
Hypotheses isubgr3stgr.v 𝑉 = ( Vtx ‘ 𝐺 )
isubgr3stgr.u 𝑈 = ( 𝐺 NeighbVtx 𝑋 )
isubgr3stgr.c 𝐶 = ( 𝐺 ClNeighbVtx 𝑋 )
isubgr3stgr.n 𝑁 ∈ ℕ0
isubgr3stgr.s 𝑆 = ( StarGr ‘ 𝑁 )
isubgr3stgr.w 𝑊 = ( Vtx ‘ 𝑆 )
isubgr3stgr.e 𝐸 = ( Edg ‘ 𝐺 )
isubgr3stgr.i 𝐼 = ( Edg ‘ ( 𝐺 ISubGr 𝐶 ) )
isubgr3stgr.h 𝐻 = ( 𝑖𝐼 ↦ ( 𝐹𝑖 ) )
Assertion isubgr3stgrlem6 ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) → 𝐻 : 𝐼 ⟶ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) )

Proof

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 = ( 𝑖𝐼 ↦ ( 𝐹𝑖 ) )
10 usgruhgr ( 𝐺 ∈ USGraph → 𝐺 ∈ UHGraph )
11 10 adantr ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) → 𝐺 ∈ UHGraph )
12 11 adantr ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) → 𝐺 ∈ UHGraph )
13 1 clnbgrssvtx ( 𝐺 ClNeighbVtx 𝑋 ) ⊆ 𝑉
14 3 13 eqsstri 𝐶𝑉
15 14 a1i ( ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) → 𝐶𝑉 )
16 eqid ( 𝐺 ISubGr 𝐶 ) = ( 𝐺 ISubGr 𝐶 )
17 1 7 16 8 isubgredg ( ( 𝐺 ∈ UHGraph ∧ 𝐶𝑉 ) → ( 𝑖𝐼 ↔ ( 𝑖𝐸𝑖𝐶 ) ) )
18 12 15 17 syl2an ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) → ( 𝑖𝐼 ↔ ( 𝑖𝐸𝑖𝐶 ) ) )
19 f1of ( 𝐹 : 𝐶1-1-onto𝑊𝐹 : 𝐶𝑊 )
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 : 𝐶1-1-onto𝑊𝑊 ⊆ ( 0 ... 𝑁 ) )
26 19 25 fssd ( 𝐹 : 𝐶1-1-onto𝑊𝐹 : 𝐶 ⟶ ( 0 ... 𝑁 ) )
27 26 ad2antrl ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) → 𝐹 : 𝐶 ⟶ ( 0 ... 𝑁 ) )
28 27 adantr ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑖𝐸𝑖𝐶 ) ) → 𝐹 : 𝐶 ⟶ ( 0 ... 𝑁 ) )
29 28 fimassd ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑖𝐸𝑖𝐶 ) ) → ( 𝐹𝑖 ) ⊆ ( 0 ... 𝑁 ) )
30 simplll ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) → 𝐺 ∈ USGraph )
31 simpl ( ( 𝑖𝐸𝑖𝐶 ) → 𝑖𝐸 )
32 1 7 usgredg ( ( 𝐺 ∈ USGraph ∧ 𝑖𝐸 ) → ∃ 𝑎𝑉𝑏𝑉 ( 𝑎𝑏𝑖 = { 𝑎 , 𝑏 } ) )
33 30 31 32 syl2an ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑖𝐸𝑖𝐶 ) ) → ∃ 𝑎𝑉𝑏𝑉 ( 𝑎𝑏𝑖 = { 𝑎 , 𝑏 } ) )
34 vex 𝑎 ∈ V
35 vex 𝑏 ∈ V
36 34 35 prss ( ( 𝑎𝐶𝑏𝐶 ) ↔ { 𝑎 , 𝑏 } ⊆ 𝐶 )
37 elclnbgrelnbgr ( ( 𝑎 ∈ ( 𝐺 ClNeighbVtx 𝑋 ) ∧ 𝑎𝑋 ) → 𝑎 ∈ ( 𝐺 NeighbVtx 𝑋 ) )
38 37 expcom ( 𝑎𝑋 → ( 𝑎 ∈ ( 𝐺 ClNeighbVtx 𝑋 ) → 𝑎 ∈ ( 𝐺 NeighbVtx 𝑋 ) ) )
39 3 eleq2i ( 𝑎𝐶𝑎 ∈ ( 𝐺 ClNeighbVtx 𝑋 ) )
40 2 eleq2i ( 𝑎𝑈𝑎 ∈ ( 𝐺 NeighbVtx 𝑋 ) )
41 38 39 40 3imtr4g ( 𝑎𝑋 → ( 𝑎𝐶𝑎𝑈 ) )
42 elclnbgrelnbgr ( ( 𝑏 ∈ ( 𝐺 ClNeighbVtx 𝑋 ) ∧ 𝑏𝑋 ) → 𝑏 ∈ ( 𝐺 NeighbVtx 𝑋 ) )
43 42 expcom ( 𝑏𝑋 → ( 𝑏 ∈ ( 𝐺 ClNeighbVtx 𝑋 ) → 𝑏 ∈ ( 𝐺 NeighbVtx 𝑋 ) ) )
44 3 eleq2i ( 𝑏𝐶𝑏 ∈ ( 𝐺 ClNeighbVtx 𝑋 ) )
45 2 eleq2i ( 𝑏𝑈𝑏 ∈ ( 𝐺 NeighbVtx 𝑋 ) )
46 43 44 45 3imtr4g ( 𝑏𝑋 → ( 𝑏𝐶𝑏𝑈 ) )
47 41 46 im2anan9r ( ( 𝑏𝑋𝑎𝑋 ) → ( ( 𝑎𝐶𝑏𝐶 ) → ( 𝑎𝑈𝑏𝑈 ) ) )
48 47 imp ( ( ( 𝑏𝑋𝑎𝑋 ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) → ( 𝑎𝑈𝑏𝑈 ) )
49 48 3adant3 ( ( ( 𝑏𝑋𝑎𝑋 ) ∧ ( 𝑎𝐶𝑏𝐶 ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) → ( 𝑎𝑈𝑏𝑈 ) )
50 preq1 ( 𝑥 = 𝑎 → { 𝑥 , 𝑦 } = { 𝑎 , 𝑦 } )
51 eqidd ( 𝑥 = 𝑎𝐸 = 𝐸 )
52 50 51 neleq12d ( 𝑥 = 𝑎 → ( { 𝑥 , 𝑦 } ∉ 𝐸 ↔ { 𝑎 , 𝑦 } ∉ 𝐸 ) )
53 preq2 ( 𝑦 = 𝑏 → { 𝑎 , 𝑦 } = { 𝑎 , 𝑏 } )
54 eqidd ( 𝑦 = 𝑏𝐸 = 𝐸 )
55 53 54 neleq12d ( 𝑦 = 𝑏 → ( { 𝑎 , 𝑦 } ∉ 𝐸 ↔ { 𝑎 , 𝑏 } ∉ 𝐸 ) )
56 52 55 rspc2v ( ( 𝑎𝑈𝑏𝑈 ) → ( ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 → { 𝑎 , 𝑏 } ∉ 𝐸 ) )
57 49 56 syl ( ( ( 𝑏𝑋𝑎𝑋 ) ∧ ( 𝑎𝐶𝑏𝐶 ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) → ( ∀ 𝑥𝑈𝑦𝑈 { 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 ( ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ( { 𝑎 , 𝑏 } ∉ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) )
60 59 3ad2ant3 ( ( ( 𝑏𝑋𝑎𝑋 ) ∧ ( 𝑎𝐶𝑏𝐶 ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) → ( { 𝑎 , 𝑏 } ∉ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) )
61 57 60 syld ( ( ( 𝑏𝑋𝑎𝑋 ) ∧ ( 𝑎𝐶𝑏𝐶 ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) → ( ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) )
62 61 3exp ( ( 𝑏𝑋𝑎𝑋 ) → ( ( 𝑎𝐶𝑏𝐶 ) → ( ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ( ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) )
63 62 com24 ( ( 𝑏𝑋𝑎𝑋 ) → ( ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 → ( ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ( ( 𝑎𝐶𝑏𝐶 ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) )
64 63 adantld ( ( 𝑏𝑋𝑎𝑋 ) → ( ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) → ( ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ( ( 𝑎𝐶𝑏𝐶 ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) )
65 64 adantld ( ( 𝑏𝑋𝑎𝑋 ) → ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) → ( ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ( ( 𝑎𝐶𝑏𝐶 ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) )
66 65 adantrd ( ( 𝑏𝑋𝑎𝑋 ) → ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) → ( ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ( ( 𝑎𝐶𝑏𝐶 ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) )
67 66 imp4c ( ( 𝑏𝑋𝑎𝑋 ) → ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) )
68 simpl ( ( 𝑏 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) ) → 𝑏 = 𝑋 )
69 simpllr ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) → ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) )
70 69 adantl ( ( 𝑏 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) ) → ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) )
71 simplrl ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) → 𝑎𝑏 )
72 71 necomd ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) → 𝑏𝑎 )
73 72 adantl ( ( 𝑏 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) ) → 𝑏𝑎 )
74 simprrr ( ( 𝑏 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) ) → 𝑏𝐶 )
75 simprrl ( ( 𝑏 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) ) → 𝑎𝐶 )
76 1 2 3 4 5 6 7 isubgr3stgrlem4 ( ( 𝑏 = 𝑋 ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ∧ ( 𝑏𝑎𝑏𝐶𝑎𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑏 , 𝑎 } ) = { 0 , 𝑧 } )
77 68 70 73 74 75 76 syl113anc ( ( 𝑏 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) ) → ∃ 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 ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } )
83 82 ex ( 𝑏 = 𝑋 → ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) )
84 simpl ( ( 𝑎 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) ) → 𝑎 = 𝑋 )
85 69 adantl ( ( 𝑎 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) ) → ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) )
86 71 adantl ( ( 𝑎 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) ) → 𝑎𝑏 )
87 simprrl ( ( 𝑎 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) ) → 𝑎𝐶 )
88 simprrr ( ( 𝑎 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) ) → 𝑏𝐶 )
89 1 2 3 4 5 6 7 isubgr3stgrlem4 ( ( 𝑎 = 𝑋 ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ∧ ( 𝑎𝑏𝑎𝐶𝑏𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } )
90 84 85 86 87 88 89 syl113anc ( ( 𝑎 = 𝑋 ∧ ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } )
91 90 ex ( 𝑎 = 𝑋 → ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) )
92 67 83 91 pm2.61iine ( ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ∧ ( 𝑎𝐶𝑏𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } )
93 92 ex ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) → ( ( 𝑎𝐶𝑏𝐶 ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) )
94 36 93 biimtrrid ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏 ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) → ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) )
95 94 exp32 ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) → ( 𝑎𝑏 → ( { 𝑎 , 𝑏 } ∈ 𝐸 → ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) )
96 95 adantrd ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) → ( ( 𝑎𝑏𝑖 = { 𝑎 , 𝑏 } ) → ( { 𝑎 , 𝑏 } ∈ 𝐸 → ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) )
97 96 imp ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏𝑖 = { 𝑎 , 𝑏 } ) ) → ( { 𝑎 , 𝑏 } ∈ 𝐸 → ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) )
98 97 com23 ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏𝑖 = { 𝑎 , 𝑏 } ) ) → ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ( { 𝑎 , 𝑏 } ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) )
99 sseq1 ( 𝑖 = { 𝑎 , 𝑏 } → ( 𝑖𝐶 ↔ { 𝑎 , 𝑏 } ⊆ 𝐶 ) )
100 eleq1 ( 𝑖 = { 𝑎 , 𝑏 } → ( 𝑖𝐸 ↔ { 𝑎 , 𝑏 } ∈ 𝐸 ) )
101 imaeq2 ( 𝑖 = { 𝑎 , 𝑏 } → ( 𝐹𝑖 ) = ( 𝐹 “ { 𝑎 , 𝑏 } ) )
102 101 eqeq1d ( 𝑖 = { 𝑎 , 𝑏 } → ( ( 𝐹𝑖 ) = { 0 , 𝑧 } ↔ ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) )
103 102 rexbidv ( 𝑖 = { 𝑎 , 𝑏 } → ( ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹𝑖 ) = { 0 , 𝑧 } ↔ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) )
104 100 103 imbi12d ( 𝑖 = { 𝑎 , 𝑏 } → ( ( 𝑖𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹𝑖 ) = { 0 , 𝑧 } ) ↔ ( { 𝑎 , 𝑏 } ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) )
105 99 104 imbi12d ( 𝑖 = { 𝑎 , 𝑏 } → ( ( 𝑖𝐶 → ( 𝑖𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹𝑖 ) = { 0 , 𝑧 } ) ) ↔ ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ( { 𝑎 , 𝑏 } ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) )
106 105 adantl ( ( 𝑎𝑏𝑖 = { 𝑎 , 𝑏 } ) → ( ( 𝑖𝐶 → ( 𝑖𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹𝑖 ) = { 0 , 𝑧 } ) ) ↔ ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ( { 𝑎 , 𝑏 } ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) )
107 106 adantl ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏𝑖 = { 𝑎 , 𝑏 } ) ) → ( ( 𝑖𝐶 → ( 𝑖𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹𝑖 ) = { 0 , 𝑧 } ) ) ↔ ( { 𝑎 , 𝑏 } ⊆ 𝐶 → ( { 𝑎 , 𝑏 } ∈ 𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑎 , 𝑏 } ) = { 0 , 𝑧 } ) ) ) )
108 98 107 mpbird ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑎𝑏𝑖 = { 𝑎 , 𝑏 } ) ) → ( 𝑖𝐶 → ( 𝑖𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹𝑖 ) = { 0 , 𝑧 } ) ) )
109 108 ex ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) → ( ( 𝑎𝑏𝑖 = { 𝑎 , 𝑏 } ) → ( 𝑖𝐶 → ( 𝑖𝐸 → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹𝑖 ) = { 0 , 𝑧 } ) ) ) )
110 109 com24 ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) → ( 𝑖𝐸 → ( 𝑖𝐶 → ( ( 𝑎𝑏𝑖 = { 𝑎 , 𝑏 } ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹𝑖 ) = { 0 , 𝑧 } ) ) ) )
111 110 imp32 ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑖𝐸𝑖𝐶 ) ) → ( ( 𝑎𝑏𝑖 = { 𝑎 , 𝑏 } ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹𝑖 ) = { 0 , 𝑧 } ) )
112 111 a1d ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑖𝐸𝑖𝐶 ) ) → ( ( 𝑎𝑉𝑏𝑉 ) → ( ( 𝑎𝑏𝑖 = { 𝑎 , 𝑏 } ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹𝑖 ) = { 0 , 𝑧 } ) ) )
113 112 rexlimdvv ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑖𝐸𝑖𝐶 ) ) → ( ∃ 𝑎𝑉𝑏𝑉 ( 𝑎𝑏𝑖 = { 𝑎 , 𝑏 } ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹𝑖 ) = { 0 , 𝑧 } ) )
114 33 113 mpd ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑖𝐸𝑖𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹𝑖 ) = { 0 , 𝑧 } )
115 stgredgel ( 𝑁 ∈ ℕ0 → ( ( 𝐹𝑖 ) ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) ↔ ( ( 𝐹𝑖 ) ⊆ ( 0 ... 𝑁 ) ∧ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹𝑖 ) = { 0 , 𝑧 } ) ) )
116 4 115 ax-mp ( ( 𝐹𝑖 ) ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) ↔ ( ( 𝐹𝑖 ) ⊆ ( 0 ... 𝑁 ) ∧ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹𝑖 ) = { 0 , 𝑧 } ) )
117 29 114 116 sylanbrc ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ ( 𝑖𝐸𝑖𝐶 ) ) → ( 𝐹𝑖 ) ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) )
118 18 117 sylbida ( ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) ∧ 𝑖𝐼 ) → ( 𝐹𝑖 ) ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) )
119 118 9 fmptd ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑋𝑉 ) ∧ ( ( ♯ ‘ 𝑈 ) = 𝑁 ∧ ∀ 𝑥𝑈𝑦𝑈 { 𝑥 , 𝑦 } ∉ 𝐸 ) ) ∧ ( 𝐹 : 𝐶1-1-onto𝑊 ∧ ( 𝐹𝑋 ) = 0 ) ) → 𝐻 : 𝐼 ⟶ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) )