Theorem fusgr2wsp2nb 30694

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 30693	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (𝑧 ∈ (𝑀‘𝑁) ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁)))
4	3	adantl 486	. . . 4 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑧 ∈ (𝑀‘𝑁) ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁)))
5	1	wspthsnwspthsnon 30274	. . . . . . 7 ⊢ (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦))
6		fusgrusgr 29681	. . . . . . . . . 10 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)
7	6	adantr 485	. . . . . . . . 9 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ USGraph)
8		eqid 2763	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
9	1, 8	usgr2wspthon 30326	. . . . . . . . 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 591	. . . . . . . 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 3228	. . . . . . 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 286	. . . . . 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 642	. . . . 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 1980	. . . . . . 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 4605	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)
16	15	bicomi 227	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨“𝑥𝑁𝑦”⟩ ↔ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})
17	16	anbi2i 634	. . . . . . . . . 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 780	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁)) → 𝑁 ∈ 𝑉)
20		anass 473	. . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . 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 657	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ (𝑥 ≠ 𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))
23		nesym 3014	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑦 = 𝑥)
24		prcom 4698	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑚, 𝑦} = {𝑦, 𝑚}
25	24	eleq1i 2854	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑚, 𝑦} ∈ (Edg‘𝐺) ↔ {𝑦, 𝑚} ∈ (Edg‘𝐺))
26	23, 25	anbi12ci 640	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ≠ 𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)) ↔ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))
27	26	anbi2i 634	. . . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
29	28	anbi1i 635	. . . . . . . . . . . . . . 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 300	. . . . . . . . . . . . . 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 4700	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑁 → {𝑥, 𝑚} = {𝑥, 𝑁})
32	31	eleq1d 2848	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑁 → ({𝑥, 𝑚} ∈ (Edg‘𝐺) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
33		preq2 4700	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑁 → {𝑦, 𝑚} = {𝑦, 𝑁})
34	33	eleq1d 2848	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑁 → ({𝑦, 𝑚} ∈ (Edg‘𝐺) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
35	34	anbi1d 642	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑁 → (({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
36	32, 35	anbi12d 643	. . . . . . . . . . . . . . 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 14912	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑁 → ⟨“𝑥𝑚𝑦”⟩ = ⟨“𝑥𝑁𝑦”⟩)
38	37	eqeq2d 2774	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑁 → (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ↔ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))
39	36, 38	anbi12d 643	. . . . . . . . . . . . . 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 286	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 6880	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = (⟨“𝑥𝑚𝑦”⟩‘1))
43		s3fv1 14934	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ V → (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚)
44	43	elv 3460	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚
45	42, 44	eqtrdi 2814	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = 𝑚)
46	45	eqeq1d 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → ((𝑧‘1) = 𝑁 ↔ 𝑚 = 𝑁))
47	46	biimpd 232	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → ((𝑧‘1) = 𝑁 → 𝑚 = 𝑁))
48	47	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) → ((𝑧‘1) = 𝑁 → 𝑚 = 𝑁))
49	48	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → ((𝑧‘1) = 𝑁 → 𝑚 = 𝑁))
50	49	com12 33	. . . . . . . . . . . . . 14 ⊢ ((𝑧‘1) = 𝑁 → (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → 𝑚 = 𝑁))
51	50	ad2antll 741	. . . . . . . . . . . . 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 411	. . . . . . . . . . . 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 3575	. . . . . . . . . . 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 589	. . . . . . . . . 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 658	. . . . . . . . . . 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 29540	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
58	1, 8	umgrpredgv 29499	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → (𝑥 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))
59	58	simpld 499	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → 𝑥 ∈ 𝑉)
60	59	ex 417	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 ∈ UMGraph → ({𝑥, 𝑁} ∈ (Edg‘𝐺) → 𝑥 ∈ 𝑉))
61	1, 8	umgrpredgv 29499	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → (𝑦 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))
62	61	simpld 499	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → 𝑦 ∈ 𝑉)
63	62	expcom 418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑦, 𝑁} ∈ (Edg‘𝐺) → (𝐺 ∈ UMGraph → 𝑦 ∈ 𝑉))
64	63	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) → (𝐺 ∈ UMGraph → 𝑦 ∈ 𝑉))
65	64	com12 33	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 ∈ UMGraph → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ 𝑉))
66	60, 65	anim12d 620	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ UMGraph → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
67	6, 57, 66	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ FinUSGraph → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
68	67	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
69	68	com12 33	. . . . . . . . . . . . . . 15 ⊢ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
70	69	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
71	70	impcom 412	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
72		fveq1 6880	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨“𝑥𝑁𝑦”⟩ → (𝑧‘1) = (⟨“𝑥𝑁𝑦”⟩‘1))
73	72	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → (𝑧‘1) = (⟨“𝑥𝑁𝑦”⟩‘1))
74		s3fv1 14934	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ 𝑉 → (⟨“𝑥𝑁𝑦”⟩‘1) = 𝑁)
75	74	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (⟨“𝑥𝑁𝑦”⟩‘1) = 𝑁)
76	73, 75	sylan9eqr 2820	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → (𝑧‘1) = 𝑁)
77	71, 76	jca 520	. . . . . . . . . . . 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 417	. . . . . . . . . . 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 571	. . . . . . . . . 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 314	. . . . . . . . 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 29712	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
82	6, 81	syl 18	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FinUSGraph → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
83	82	adantr 485	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
84		eldif 3915	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}))
85	8	nbusgreledg 29712	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ USGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
86	6, 85	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ FinUSGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
87	86	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
88		velsn 4605	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
89	88	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥))
90	89	notbid 321	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (¬ 𝑦 ∈ {𝑥} ↔ ¬ 𝑦 = 𝑥))
91	87, 90	anbi12d 643	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → ((𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
92	84, 91	bitrid 286	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
93	83, 92	anbi12d 643	. . . . . . . . . 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 642	. . . . . . . . 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 314	. . . . . . . 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 1954	. . . . . . 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 288	. . . . . 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 3202	. . . . . . 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 635	. . . . . 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 3202	. . . . . 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 317	. . . . 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 3459	. . . . . . . 8 ⊢ 𝑧 ∈ V
103		eleq1w 2846	. . . . . . . . 9 ⊢ (𝑝 = 𝑧 → (𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
104	103	2rexbidv 3230	. . . . . . . 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 3638	. . . . . . 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 227	. . . . . 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 308	. . . 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 282	. . 3 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑧 ∈ (𝑀‘𝑁) ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}))
110	109	eqrdv 2761	. 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑀‘𝑁) = {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}})
111		dfiunv2 4998	. 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 2816	1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑀‘𝑁) = ∪ 𝑥 ∈ (𝐺 NeighbVtx 𝑁)∪ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){⟨“𝑥𝑁𝑦”⟩})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741   ≠ wne 2958  ∃wrex 3089  {crab 3416  Vcvv 3455   ∖ cdif 3902  {csn 4589  {cpr 4591  ∪ ciun 4956   ↦ cmpt 5192  ‘cfv 6536  (class class class)co 7410  1c1 11105  2c2 12299  ⟨“cs3 14884  Vtxcvtx 29355  Edgcedg 29406  UMGraphcumgr 29440  USGraphcusgr 29508  FinUSGraphcfusgr 29675   NeighbVtx cnbgr 29691   WSPathsN cwwspthsn 30186   WSPathsNOn cwwspthsnon 30187

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-oadd 8453  df-er 8690  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9892  df-card 9930  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-nn 12238  df-2 12307  df-3 12308  df-n0 12509  df-xnn0 12582  df-z 12596  df-uz 12867  df-fz 13540  df-fzo 13688  df-hash 14372  df-word 14556  df-concat 14613  df-s1 14639  df-s2 14890  df-s3 14891  df-edg 29407  df-uhgr 29417  df-upgr 29441  df-umgr 29442  df-uspgr 29509  df-usgr 29510  df-fusgr 29676  df-nbgr 29692  df-wlks 29958  df-wlkson 29959  df-trls 30049  df-trlson 30050  df-pths 30072  df-spths 30073  df-pthson 30074  df-spthson 30075  df-wwlks 30188  df-wwlksn 30189  df-wwlksnon 30190  df-wspthsn 30191  df-wspthsnon 30192

This theorem is used by:  fusgreghash2wspv  30695