Theorem fusgr2wsp2nb 30362

Description: The set of paths of length 2 with a given vertex in the middle for a finite simple graph is the union of all paths of length 2 from one neighbor to another neighbor of this vertex via this vertex. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 16-Mar-2022.)

Hypotheses

Ref	Expression
frgrhash2wsp.v	⊢ 𝑉 = (Vtx‘𝐺)
fusgreg2wsp.m	⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})

Assertion

Ref	Expression
fusgr2wsp2nb	⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑀‘𝑁) = ∪ 𝑥 ∈ (𝐺 NeighbVtx 𝑁)∪ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){⟨“𝑥𝑁𝑦”⟩})

Distinct variable groups:   𝐺,𝑎   𝑉,𝑎   𝑤,𝐺   𝑁,𝑎,𝑤   𝑥,𝐺,𝑦   𝑥,𝑁,𝑦   𝑥,𝑉,𝑦

Allowed substitution hints:   𝑀(𝑥,𝑦,𝑤,𝑎)   𝑉(𝑤)

Proof of Theorem fusgr2wsp2nb

Dummy variables 𝑚 𝑧 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrhash2wsp.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
2		fusgreg2wsp.m	. . . . . 6 ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
3	1, 2	fusgreg2wsplem 30361	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (𝑧 ∈ (𝑀‘𝑁) ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁)))
4	3	adantl 481	. . . 4 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑧 ∈ (𝑀‘𝑁) ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁)))
5	1	wspthsnwspthsnon 29945	. . . . . . 7 ⊢ (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦))
6		fusgrusgr 29353	. . . . . . . . . 10 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)
7	6	adantr 480	. . . . . . . . 9 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ USGraph)
8		eqid 2734	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
9	1, 8	usgr2wspthon 29994	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
10	7, 9	sylan 580	. . . . . . . 8 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
11	10	2rexbidva 3217	. . . . . . 7 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
12	5, 11	bitrid 283	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
13	12	anbi1d 631	. . . . 5 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁) ↔ (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ (𝑧‘1) = 𝑁)))
14		19.41vv 1947	. . . . . . 7 ⊢ (∃𝑥∃𝑦(((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ (∃𝑥∃𝑦((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁))
15		velsn 4646	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)
16	15	bicomi 224	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨“𝑥𝑁𝑦”⟩ ↔ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})
17	16	anbi2i 623	. . . . . . . . . 10 ⊢ ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
18	17	a1i 11	. . . . . . . . 9 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})))
19		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁)) → 𝑁 ∈ 𝑉)
20		anass 468	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ (𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
21		ancom 460	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ (𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ↔ ((𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ 𝑧 = ⟨“𝑥𝑚𝑦”⟩))
22		an12 645	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ (𝑥 ≠ 𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))
23		nesym 2994	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑦 = 𝑥)
24		prcom 4736	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑚, 𝑦} = {𝑦, 𝑚}
25	24	eleq1i 2829	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑚, 𝑦} ∈ (Edg‘𝐺) ↔ {𝑦, 𝑚} ∈ (Edg‘𝐺))
26	23, 25	anbi12ci 629	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ≠ 𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)) ↔ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))
27	26	anbi2i 623	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ (𝑥 ≠ 𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
28	22, 27	bitri 275	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
29	28	anbi1i 624	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ 𝑧 = ⟨“𝑥𝑚𝑦”⟩) ↔ (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑚𝑦”⟩))
30	20, 21, 29	3bitri 297	. . . . . . . . . . . . . 14 ⊢ (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑚𝑦”⟩))
31		preq2 4738	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑁 → {𝑥, 𝑚} = {𝑥, 𝑁})
32	31	eleq1d 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑁 → ({𝑥, 𝑚} ∈ (Edg‘𝐺) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
33		preq2 4738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑁 → {𝑦, 𝑚} = {𝑦, 𝑁})
34	33	eleq1d 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑁 → ({𝑦, 𝑚} ∈ (Edg‘𝐺) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
35	34	anbi1d 631	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑁 → (({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
36	32, 35	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑁 → (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ↔ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))))
37		s3eq2 14905	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑁 → ⟨“𝑥𝑚𝑦”⟩ = ⟨“𝑥𝑁𝑦”⟩)
38	37	eqeq2d 2745	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑁 → (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ↔ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))
39	36, 38	anbi12d 632	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑁 → ((({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑚𝑦”⟩) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))
40	30, 39	bitrid 283	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑁 → (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))
41	40	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁)) ∧ 𝑚 = 𝑁) → (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))
42		fveq1 6905	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = (⟨“𝑥𝑚𝑦”⟩‘1))
43		s3fv1 14927	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ V → (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚)
44	43	elv 3482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚
45	42, 44	eqtrdi 2790	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = 𝑚)
46	45	eqeq1d 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → ((𝑧‘1) = 𝑁 ↔ 𝑚 = 𝑁))
47	46	biimpd 229	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → ((𝑧‘1) = 𝑁 → 𝑚 = 𝑁))
48	47	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) → ((𝑧‘1) = 𝑁 → 𝑚 = 𝑁))
49	48	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → ((𝑧‘1) = 𝑁 → 𝑚 = 𝑁))
50	49	com12 32	. . . . . . . . . . . . . 14 ⊢ ((𝑧‘1) = 𝑁 → (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → 𝑚 = 𝑁))
51	50	ad2antll 729	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁)) → (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → 𝑚 = 𝑁))
52	51	imp 406	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁)) ∧ ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) → 𝑚 = 𝑁)
53	19, 41, 52	rspcebdv 3615	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁)) → (∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))
54	53	pm5.32da 579	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ↔ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))))
55		an32 646	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
56	55	a1i 11	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))))
57		usgrumgr 29212	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
58	1, 8	umgrpredgv 29171	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → (𝑥 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))
59	58	simpld 494	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → 𝑥 ∈ 𝑉)
60	59	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 ∈ UMGraph → ({𝑥, 𝑁} ∈ (Edg‘𝐺) → 𝑥 ∈ 𝑉))
61	1, 8	umgrpredgv 29171	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → (𝑦 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))
62	61	simpld 494	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → 𝑦 ∈ 𝑉)
63	62	expcom 413	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑦, 𝑁} ∈ (Edg‘𝐺) → (𝐺 ∈ UMGraph → 𝑦 ∈ 𝑉))
64	63	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) → (𝐺 ∈ UMGraph → 𝑦 ∈ 𝑉))
65	64	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 ∈ UMGraph → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ 𝑉))
66	60, 65	anim12d 609	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ UMGraph → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
67	6, 57, 66	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ FinUSGraph → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
68	67	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
69	68	com12 32	. . . . . . . . . . . . . . 15 ⊢ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
70	69	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
71	70	impcom 407	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
72		fveq1 6905	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨“𝑥𝑁𝑦”⟩ → (𝑧‘1) = (⟨“𝑥𝑁𝑦”⟩‘1))
73	72	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → (𝑧‘1) = (⟨“𝑥𝑁𝑦”⟩‘1))
74		s3fv1 14927	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ 𝑉 → (⟨“𝑥𝑁𝑦”⟩‘1) = 𝑁)
75	74	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (⟨“𝑥𝑁𝑦”⟩‘1) = 𝑁)
76	73, 75	sylan9eqr 2796	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → (𝑧‘1) = 𝑁)
77	71, 76	jca 511	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁))
78	77	ex 412	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁)))
79	78	pm4.71rd 562	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) ↔ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))))
80	54, 56, 79	3bitr4d 311	. . . . . . . . 9 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))
81	8	nbusgreledg 29384	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
82	6, 81	syl 17	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FinUSGraph → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
83	82	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
84		eldif 3972	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}))
85	8	nbusgreledg 29384	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
86	6, 85	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ FinUSGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
87	86	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
88		velsn 4646	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
89	88	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥))
90	89	notbid 318	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (¬ 𝑦 ∈ {𝑥} ↔ ¬ 𝑦 = 𝑥))
91	87, 90	anbi12d 632	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
92	84, 91	bitrid 283	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
93	83, 92	anbi12d 632	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ↔ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))))
94	93	anbi1d 631	. . . . . . . . 9 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})))
95	18, 80, 94	3bitr4d 311	. . . . . . . 8 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ ((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})))
96	95	2exbidv 1921	. . . . . . 7 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑥∃𝑦(((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ ∃𝑥∃𝑦((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})))
97	14, 96	bitr3id 285	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((∃𝑥∃𝑦((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ ∃𝑥∃𝑦((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})))
98		r2ex 3193	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
99	98	anbi1i 624	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ (𝑧‘1) = 𝑁) ↔ (∃𝑥∃𝑦((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁))
100		r2ex 3193	. . . . . 6 ⊢ (∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ ∃𝑥∃𝑦((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
101	97, 99, 100	3bitr4g 314	. . . . 5 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑚 ∈ 𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ (𝑧‘1) = 𝑁) ↔ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
102		vex 3481	. . . . . . . 8 ⊢ 𝑧 ∈ V
103		eleq1w 2821	. . . . . . . . 9 ⊢ (𝑝 = 𝑧 → (𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
104	103	2rexbidv 3219	. . . . . . . 8 ⊢ (𝑝 = 𝑧 → (∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
105	102, 104	elab 3680	. . . . . . 7 ⊢ (𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}} ↔ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})
106	105	bicomi 224	. . . . . 6 ⊢ (∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}})
107	106	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}))
108	13, 101, 107	3bitrd 305	. . . 4 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁) ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}))
109	4, 108	bitrd 279	. . 3 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑧 ∈ (𝑀‘𝑁) ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}))
110	109	eqrdv 2732	. 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑀‘𝑁) = {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}})
111		dfiunv2 5039	. 2 ⊢ ∪ 𝑥 ∈ (𝐺 NeighbVtx 𝑁)∪ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){⟨“𝑥𝑁𝑦”⟩} = {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}
112	110, 111	eqtr4di 2792	1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑀‘𝑁) = ∪ 𝑥 ∈ (𝐺 NeighbVtx 𝑁)∪ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){⟨“𝑥𝑁𝑦”⟩})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536  ∃wex 1775   ∈ wcel 2105  {cab 2711   ≠ wne 2937  ∃wrex 3067  {crab 3432  Vcvv 3477   ∖ cdif 3959  {csn 4630  {cpr 4632  ∪ ciun 4995   ↦ cmpt 5230  ‘cfv 6562  (class class class)co 7430  1c1 11153  2c2 12318  ⟨“cs3 14877  Vtxcvtx 29027  Edgcedg 29078  UMGraphcumgr 29112  USGraphcusgr 29180  FinUSGraphcfusgr 29347   NeighbVtx cnbgr 29363   WSPathsN cwwspthsn 29857   WSPathsNOn cwwspthsnon 29858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-ac2 10500  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-ac 10153  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-hash 14366  df-word 14549  df-concat 14605  df-s1 14630  df-s2 14883  df-s3 14884  df-edg 29079  df-uhgr 29089  df-upgr 29113  df-umgr 29114  df-uspgr 29181  df-usgr 29182  df-fusgr 29348  df-nbgr 29364  df-wlks 29631  df-wlkson 29632  df-trls 29724  df-trlson 29725  df-pths 29748  df-spths 29749  df-pthson 29750  df-spthson 29751  df-wwlks 29859  df-wwlksn 29860  df-wwlksnon 29861  df-wspthsn 29862  df-wspthsnon 29863

This theorem is referenced by:  fusgreghash2wspv  30363