isubgr3stgrlem6 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > isubgr3stgrlem6		Structured version Visualization version GIF version

Theorem isubgr3stgrlem6 47931

Description: Lemma 6 for isubgr3stgr 47935. (Contributed by AV, 24-Sep-2025.)

**Hypotheses**
Ref	Expression
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**
Ref	Expression
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)))

Distinct variable groups: 𝐶,𝑖 𝑖,𝐹 𝑖,𝐼 𝑖,𝑊 𝑖,𝐸,𝑥,𝑦 𝑖,𝐺 𝑖,𝑁 𝑈,𝑖,𝑥,𝑦 𝑖,𝑉 𝑖,𝑋 Allowed substitution hints: 𝐶(𝑥,𝑦) 𝑆(𝑥,𝑦,𝑖) 𝐹(𝑥,𝑦) 𝐺(𝑥,𝑦) 𝐻(𝑥,𝑦,𝑖) 𝐼(𝑥,𝑦) 𝑁(𝑥,𝑦) 𝑉(𝑥,𝑦) 𝑊(𝑥,𝑦) 𝑋(𝑥,𝑦)

Proof of Theorem isubgr3stgrlem6 Dummy variables 𝑧 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgruhgr 29111	. . . . . 6 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
2	1	adantr 480	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ UHGraph)
3	2	adantr 480	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) → 𝐺 ∈ UHGraph)
4		isubgr3stgr.c	. . . . . 6 ⊢ 𝐶 = (𝐺 ClNeighbVtx 𝑋)
5		isubgr3stgr.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
6	5	clnbgrssvtx 47793	. . . . . 6 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
7	4, 6	eqsstri 4005	. . . . 5 ⊢ 𝐶 ⊆ 𝑉
8	7	a1i 11	. . . 4 ⊢ ((𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0) → 𝐶 ⊆ 𝑉)
9		isubgr3stgr.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
10		eqid 2735	. . . . 5 ⊢ (𝐺 ISubGr 𝐶) = (𝐺 ISubGr 𝐶)
11		isubgr3stgr.i	. . . . 5 ⊢ 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))
12	5, 9, 10, 11	isubgredg 47827	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐶 ⊆ 𝑉) → (𝑖 ∈ 𝐼 ↔ (𝑖 ∈ 𝐸 ∧ 𝑖 ⊆ 𝐶)))
13	3, 8, 12	syl2an 596	. . 3 ⊢ ((((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)))
14		f1of 6817	. . . . . . . 8 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝐹:𝐶⟶𝑊)
15		isubgr3stgr.w	. . . . . . . . . . 11 ⊢ 𝑊 = (Vtx‘𝑆)
16		isubgr3stgr.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (StarGr‘𝑁)
17	16	fveq2i 6878	. . . . . . . . . . 11 ⊢ (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
18		isubgr3stgr.n	. . . . . . . . . . . 12 ⊢ 𝑁 ∈ ℕ₀
19		stgrvtx 47914	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ → (Vtx‘(StarGr‘𝑁)) = (0...𝑁))
20	18, 19	ax-mp 5	. . . . . . . . . . 11 ⊢ (Vtx‘(StarGr‘𝑁)) = (0...𝑁)
21	15, 17, 20	3eqtri 2762	. . . . . . . . . 10 ⊢ 𝑊 = (0...𝑁)
22	21	eqimssi 4019	. . . . . . . . 9 ⊢ 𝑊 ⊆ (0...𝑁)
23	22	a1i 11	. . . . . . . 8 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝑊 ⊆ (0...𝑁))
24	14, 23	fssd 6722	. . . . . . 7 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝐹:𝐶⟶(0...𝑁))
25	24	ad2antrl 728	. . . . . 6 ⊢ ((((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))
26	25	adantr 480	. . . . 5 ⊢ (((((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))
27	26	fimassd 6726	. . . 4 ⊢ (((((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))
28		simplll 774	. . . . . 6 ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶–1-1-onto→𝑊 ∧ (𝐹‘𝑋) = 0)) → 𝐺 ∈ USGraph)
29		simpl 482	. . . . . 6 ⊢ ((𝑖 ∈ 𝐸 ∧ 𝑖 ⊆ 𝐶) → 𝑖 ∈ 𝐸)
30	5, 9	usgredg 29124	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑖 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑖 = {𝑎, 𝑏}))
31	28, 29, 30	syl2an 596	. . . . 5 ⊢ (((((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}))
32		vex 3463	. . . . . . . . . . . . . . . . 17 ⊢ 𝑎 ∈ V
33		vex 3463	. . . . . . . . . . . . . . . . 17 ⊢ 𝑏 ∈ V
34	32, 33	prss 4796	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) ↔ {𝑎, 𝑏} ⊆ 𝐶)
35		elclnbgrelnbgr 47787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑎 ∈ (𝐺 ClNeighbVtx 𝑋) ∧ 𝑎 ≠ 𝑋) → 𝑎 ∈ (𝐺 NeighbVtx 𝑋))
36	35	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 ≠ 𝑋 → (𝑎 ∈ (𝐺 ClNeighbVtx 𝑋) → 𝑎 ∈ (𝐺 NeighbVtx 𝑋)))
37	4	eleq2i 2826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 ∈ 𝐶 ↔ 𝑎 ∈ (𝐺 ClNeighbVtx 𝑋))
38		isubgr3stgr.u	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
39	38	eleq2i 2826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 ∈ 𝑈 ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑋))
40	36, 37, 39	3imtr4g 296	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 ≠ 𝑋 → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝑈))
41		elclnbgrelnbgr 47787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑏 ∈ (𝐺 ClNeighbVtx 𝑋) ∧ 𝑏 ≠ 𝑋) → 𝑏 ∈ (𝐺 NeighbVtx 𝑋))
42	41	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 ≠ 𝑋 → (𝑏 ∈ (𝐺 ClNeighbVtx 𝑋) → 𝑏 ∈ (𝐺 NeighbVtx 𝑋)))
43	4	eleq2i 2826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 ∈ 𝐶 ↔ 𝑏 ∈ (𝐺 ClNeighbVtx 𝑋))
44	38	eleq2i 2826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 ∈ 𝑈 ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))
45	42, 43, 44	3imtr4g 296	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑏 ≠ 𝑋 → (𝑏 ∈ 𝐶 → 𝑏 ∈ 𝑈))
46	40, 45	im2anan9r 621	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋) → ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)))
47	46	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋) ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈))
48	47	3adant3 1132	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋) ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) ∧ (𝑎 ≠ 𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) → (𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈))
49		preq1 4709	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 𝑎 → {𝑥, 𝑦} = {𝑎, 𝑦})
50		eqidd 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 𝑎 → 𝐸 = 𝐸)
51	49, 50	neleq12d 3041	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = 𝑎 → ({𝑥, 𝑦} ∉ 𝐸 ↔ {𝑎, 𝑦} ∉ 𝐸))
52		preq2 4710	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = 𝑏 → {𝑎, 𝑦} = {𝑎, 𝑏})
53		eqidd 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = 𝑏 → 𝐸 = 𝐸)
54	52, 53	neleq12d 3041	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = 𝑏 → ({𝑎, 𝑦} ∉ 𝐸 ↔ {𝑎, 𝑏} ∉ 𝐸))
55	51, 54	rspc2v 3612	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈) → (∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸 → {𝑎, 𝑏} ∉ 𝐸))
56	48, 55	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋) ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) ∧ (𝑎 ≠ 𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) → (∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸 → {𝑎, 𝑏} ∉ 𝐸))
57		pm2.24nel 3049	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({𝑎, 𝑏} ∈ 𝐸 → ({𝑎, 𝑏} ∉ 𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
58	57	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑎 ≠ 𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸) → ({𝑎, 𝑏} ∉ 𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
59	58	3ad2ant3 1135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑏 ≠ 𝑋 ∧ 𝑎 ≠ 𝑋) ∧ (𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) ∧ (𝑎 ≠ 𝑏 ∧ {𝑎, 𝑏} ∈ 𝐸)) → ({𝑎, 𝑏} ∉ 𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
60	56, 59	syld 47	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((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}))
61	60	3exp 1119	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((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	com24 95	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((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}))))
63	62	adantld 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((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}))))
64	63	adantld 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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}))))
65	64	adantrd 491	. . . . . . . . . . . . . . . . . . 19 ⊢ ((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}))))
66	65	imp4c 423	. . . . . . . . . . . . . . . . . 18 ⊢ ((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		simpl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((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)
68		simpllr 775	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((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))
69	68	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((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))
70		simplrl 776	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((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)
71	70	necomd 2987	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((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)
72	71	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((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)
73		simprrr 781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((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)
74		simprrl 780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((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)
75	5, 38, 4, 18, 16, 15, 9	isubgr3stgrlem4 47929	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((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})
76	67, 69, 72, 73, 74, 75	syl113anc 1384	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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})
77		prcom 4708	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑎, 𝑏} = {𝑏, 𝑎}
78	77	imaeq2i 6045	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 “ {𝑎, 𝑏}) = (𝐹 “ {𝑏, 𝑎})
79	78	eqeq1i 2740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 “ {𝑎, 𝑏}) = {0, 𝑧} ↔ (𝐹 “ {𝑏, 𝑎}) = {0, 𝑧})
80	79	rexbii 3083	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧} ↔ ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑏, 𝑎}) = {0, 𝑧})
81	76, 80	sylibr 234	. . . . . . . . . . . . . . . . . . 19 ⊢ ((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})
82	81	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ (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		simpl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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)
84	68	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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))
85	70	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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)
86		simprrl 780	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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)
87		simprrr 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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)
88	5, 38, 4, 18, 16, 15, 9	isubgr3stgrlem4 47929	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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})
89	83, 84, 85, 86, 87, 88	syl113anc 1384	. . . . . . . . . . . . . . . . . . 19 ⊢ ((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})
90	89	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ (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	66, 82, 90	pm2.61iine 3022	. . . . . . . . . . . . . . . . 17 ⊢ ((((((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	91	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (((((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	34, 92	biimtrrid 243	. . . . . . . . . . . . . . 15 ⊢ (((((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}))
94	93	exp32 420	. . . . . . . . . . . . . 14 ⊢ ((((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	adantrd 491	. . . . . . . . . . . . 13 ⊢ ((((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}))))
96	95	imp 406	. . . . . . . . . . . 12 ⊢ (((((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	com23 86	. . . . . . . . . . 11 ⊢ (((((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})))
98		sseq1 3984	. . . . . . . . . . . . . 14 ⊢ (𝑖 = {𝑎, 𝑏} → (𝑖 ⊆ 𝐶 ↔ {𝑎, 𝑏} ⊆ 𝐶))
99		eleq1 2822	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = {𝑎, 𝑏} → (𝑖 ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
100		imaeq2 6043	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = {𝑎, 𝑏} → (𝐹 “ 𝑖) = (𝐹 “ {𝑎, 𝑏}))
101	100	eqeq1d 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = {𝑎, 𝑏} → ((𝐹 “ 𝑖) = {0, 𝑧} ↔ (𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
102	101	rexbidv 3164	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = {𝑎, 𝑏} → (∃𝑧 ∈ (1...𝑁)(𝐹 “ 𝑖) = {0, 𝑧} ↔ ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
103	99, 102	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑖 = {𝑎, 𝑏} → ((𝑖 ∈ 𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ 𝑖) = {0, 𝑧}) ↔ ({𝑎, 𝑏} ∈ 𝐸 → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑎, 𝑏}) = {0, 𝑧})))
104	98, 103	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (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}))))
105	104	adantl 481	. . . . . . . . . . . 12 ⊢ ((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}))))
106	105	adantl 481	. . . . . . . . . . 11 ⊢ (((((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}))))
107	97, 106	mpbird 257	. . . . . . . . . 10 ⊢ (((((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})))
108	107	ex 412	. . . . . . . . 9 ⊢ ((((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	com24 95	. . . . . . . 8 ⊢ ((((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}))))
110	109	imp32 418	. . . . . . 7 ⊢ (((((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	a1d 25	. . . . . 6 ⊢ (((((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})))
112	111	rexlimdvv 3197	. . . . 5 ⊢ (((((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	31, 112	mpd 15	. . . 4 ⊢ (((((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})
114		stgredgel 47917	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → ((𝐹 “ 𝑖) ∈ (Edg‘(StarGr‘𝑁)) ↔ ((𝐹 “ 𝑖) ⊆ (0...𝑁) ∧ ∃𝑧 ∈ (1...𝑁)(𝐹 “ 𝑖) = {0, 𝑧})))
115	18, 114	ax-mp 5	. . . 4 ⊢ ((𝐹 “ 𝑖) ∈ (Edg‘(StarGr‘𝑁)) ↔ ((𝐹 “ 𝑖) ⊆ (0...𝑁) ∧ ∃𝑧 ∈ (1...𝑁)(𝐹 “ 𝑖) = {0, 𝑧}))
116	27, 113, 115	sylanbrc 583	. . 3 ⊢ (((((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)))
117	13, 116	sylbida 592	. 2 ⊢ (((((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)))
118		isubgr3stgr.h	. 2 ⊢ 𝐻 = (𝑖 ∈ 𝐼 ↦ (𝐹 “ 𝑖))
119	117, 118	fmptd 7103	1 ⊢ ((((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)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∉ wnel 3036 ∀wral 3051 ∃wrex 3060 ⊆ wss 3926 {cpr 4603 ↦ cmpt 5201 “ cima 5657 ⟶wf 6526 –1-1-onto→wf1o 6529 ‘cfv 6530 (class class class)co 7403 0cc0 11127 1c1 11128 ℕ₀cn0 12499 ...cfz 13522 ♯chash 14346 Vtxcvtx 28921 Edgcedg 28972 UHGraphcuhgr 28981 USGraphcusgr 29074 NeighbVtx cnbgr 29257 ClNeighbVtx cclnbgr 47780 ISubGr cisubgr 47821 StarGrcstgr 47911

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-er 8717 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-dju 9913 df-card 9951 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12851 df-fz 13523 df-hash 14347 df-struct 17164 df-slot 17199 df-ndx 17211 df-base 17227 df-edgf 28914 df-vtx 28923 df-iedg 28924 df-edg 28973 df-uhgr 28983 df-upgr 29007 df-umgr 29008 df-uspgr 29075 df-usgr 29076 df-nbgr 29258 df-clnbgr 47781 df-isubgr 47822 df-stgr 47912

This theorem is referenced by: isubgr3stgrlem8 47933

W3C validator