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 = ( 𝑖 ∈ 𝐼 ↦ ( 𝐹 “ 𝑖 ) )
Assertion isubgr3stgrlem6 ( ( ( ( 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 ) ) )

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 = ( 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 ) ) )