isubgr3stgrlem6 - Mathbox for Alexander van der Vekens

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Theorem isubgr3stgrlem6 47963

Description: Lemma 6 for isubgr3stgr 47967. (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 29166	. . . . . 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 47825	. . . . . 6 ⊢ (𝐺 ClNeighbVtx 𝑋) ⊆ 𝑉
7	4, 6	eqsstri 3990	. . . . 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 2729	. . . . 5 ⊢ (𝐺 ISubGr 𝐶) = (𝐺 ISubGr 𝐶)
11		isubgr3stgr.i	. . . . 5 ⊢ 𝐼 = (Edg‘(𝐺 ISubGr 𝐶))
12	5, 9, 10, 11	isubgredg 47859	. . . 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 6782	. . . . . . . 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 6843	. . . . . . . . . . 11 ⊢ (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
18		isubgr3stgr.n	. . . . . . . . . . . 12 ⊢ 𝑁 ∈ ℕ₀
19		stgrvtx 47946	. . . . . . . . . . . 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 2756	. . . . . . . . . 10 ⊢ 𝑊 = (0...𝑁)
22	21	eqimssi 4004	. . . . . . . . 9 ⊢ 𝑊 ⊆ (0...𝑁)
23	22	a1i 11	. . . . . . . 8 ⊢ (𝐹:𝐶–1-1-onto→𝑊 → 𝑊 ⊆ (0...𝑁))
24	14, 23	fssd 6687	. . . . . . 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 6691	. . . 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 29179	. . . . . 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 3448	. . . . . . . . . . . . . . . . 17 ⊢ 𝑎 ∈ V
33		vex 3448	. . . . . . . . . . . . . . . . 17 ⊢ 𝑏 ∈ V
34	32, 33	prss 4780	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) ↔ {𝑎, 𝑏} ⊆ 𝐶)
35		elclnbgrelnbgr 47819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 ∈ 𝐶 ↔ 𝑎 ∈ (𝐺 ClNeighbVtx 𝑋))
38		isubgr3stgr.u	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋)
39	38	eleq2i 2820	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑎 ∈ 𝑈 ↔ 𝑎 ∈ (𝐺 NeighbVtx 𝑋))
40	36, 37, 39	3imtr4g 296	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑎 ≠ 𝑋 → (𝑎 ∈ 𝐶 → 𝑎 ∈ 𝑈))
41		elclnbgrelnbgr 47819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 ∈ 𝐶 ↔ 𝑏 ∈ (𝐺 ClNeighbVtx 𝑋))
44	38	eleq2i 2820	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4693	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 𝑎 → {𝑥, 𝑦} = {𝑎, 𝑦})
50		eqidd 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 𝑎 → 𝐸 = 𝐸)
51	49, 50	neleq12d 3034	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = 𝑎 → ({𝑥, 𝑦} ∉ 𝐸 ↔ {𝑎, 𝑦} ∉ 𝐸))
52		preq2 4694	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = 𝑏 → {𝑎, 𝑦} = {𝑎, 𝑏})
53		eqidd 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = 𝑏 → 𝐸 = 𝐸)
54	52, 53	neleq12d 3034	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = 𝑏 → ({𝑎, 𝑦} ∉ 𝐸 ↔ {𝑎, 𝑏} ∉ 𝐸))
55	51, 54	rspc2v 3596	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3042	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2980	. . . . . . . . . . . . . . . . . . . . . 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 47961	. . . . . . . . . . . . . . . . . . . . 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 4692	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑎, 𝑏} = {𝑏, 𝑎}
78	77	imaeq2i 6018	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 “ {𝑎, 𝑏}) = (𝐹 “ {𝑏, 𝑎})
79	78	eqeq1i 2734	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 “ {𝑎, 𝑏}) = {0, 𝑧} ↔ (𝐹 “ {𝑏, 𝑎}) = {0, 𝑧})
80	79	rexbii 3076	. . . . . . . . . . . . . . . . . . . 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 47961	. . . . . . . . . . . . . . . . . . . 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 3015	. . . . . . . . . . . . . . . . 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 3969	. . . . . . . . . . . . . 14 ⊢ (𝑖 = {𝑎, 𝑏} → (𝑖 ⊆ 𝐶 ↔ {𝑎, 𝑏} ⊆ 𝐶))
99		eleq1 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = {𝑎, 𝑏} → (𝑖 ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
100		imaeq2 6016	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = {𝑎, 𝑏} → (𝐹 “ 𝑖) = (𝐹 “ {𝑎, 𝑏}))
101	100	eqeq1d 2731	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = {𝑎, 𝑏} → ((𝐹 “ 𝑖) = {0, 𝑧} ↔ (𝐹 “ {𝑎, 𝑏}) = {0, 𝑧}))
102	101	rexbidv 3157	. . . . . . . . . . . . . . 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 3191	. . . . 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 47949	. . . . 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 7068	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 2109 ≠ wne 2925 ∉ wnel 3029 ∀wral 3044 ∃wrex 3053 ⊆ wss 3911 {cpr 4587 ↦ cmpt 5183 “ cima 5634 ⟶wf 6495 –1-1-onto→wf1o 6498 ‘cfv 6499 (class class class)co 7369 0cc0 11044 1c1 11045 ℕ₀cn0 12418 ...cfz 13444 ♯chash 14271 Vtxcvtx 28976 Edgcedg 29027 UHGraphcuhgr 29036 USGraphcusgr 29129 NeighbVtx cnbgr 29312 ClNeighbVtx cclnbgr 47812 ISubGr cisubgr 47853 StarGrcstgr 47943

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-fz 13445 df-hash 14272 df-struct 17093 df-slot 17128 df-ndx 17140 df-base 17156 df-edgf 28969 df-vtx 28978 df-iedg 28979 df-edg 29028 df-uhgr 29038 df-upgr 29062 df-umgr 29063 df-uspgr 29130 df-usgr 29131 df-nbgr 29313 df-clnbgr 47813 df-isubgr 47854 df-stgr 47944

This theorem is referenced by: isubgr3stgrlem8 47965

W3C validator