Theorem fusgr2wsp2nb 30263

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 30262	. . . . 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 29846	. . . . . . 7 ⊢ (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦))
6		fusgrusgr 29249	. . . . . . . . . 10 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)
7	6	adantr 480	. . . . . . . . 9 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ USGraph)
8		eqid 2729	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
9	1, 8	usgr2wspthon 29895	. . . . . . . . 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 3200	. . . . . . 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 1950	. . . . . . 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 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 768	. . . . . . . . . . . 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 2981	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑦 = 𝑥)
24		prcom 4696	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑚, 𝑦} = {𝑦, 𝑚}
25	24	eleq1i 2819	. . . . . . . . . . . . . . . . . . 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 4698	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑁 → {𝑥, 𝑚} = {𝑥, 𝑁})
32	31	eleq1d 2813	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑁 → ({𝑥, 𝑚} ∈ (Edg‘𝐺) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
33		preq2 4698	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑁 → {𝑦, 𝑚} = {𝑦, 𝑁})
34	33	eleq1d 2813	. . . . . . . . . . . . . . . . 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 14836	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑁 → ⟨“𝑥𝑚𝑦”⟩ = ⟨“𝑥𝑁𝑦”⟩)
38	37	eqeq2d 2740	. . . . . . . . . . . . . . 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 6857	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = (⟨“𝑥𝑚𝑦”⟩‘1))
43		s3fv1 14858	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ V → (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚)
44	43	elv 3452	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚
45	42, 44	eqtrdi 2780	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = 𝑚)
46	45	eqeq1d 2731	. . . . . . . . . . . . . . . . . 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 3582	. . . . . . . . . . 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 29108	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
58	1, 8	umgrpredgv 29067	. . . . . . . . . . . . . . . . . . . . 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 29067	. . . . . . . . . . . . . . . . . . . . . . 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 6857	. . . . . . . . . . . . . . 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 14858	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ 𝑉 → (⟨“𝑥𝑁𝑦”⟩‘1) = 𝑁)
75	74	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (⟨“𝑥𝑁𝑦”⟩‘1) = 𝑁)
76	73, 75	sylan9eqr 2786	. . . . . . . . . . . . 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 29280	. . . . . . . . . . . . 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 3924	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}))
85	8	nbusgreledg 29280	. . . . . . . . . . . . . . 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 4605	. . . . . . . . . . . . . . 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 1924	. . . . . . 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 3174	. . . . . . 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 3174	. . . . . 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 3451	. . . . . . . 8 ⊢ 𝑧 ∈ V
103		eleq1w 2811	. . . . . . . . 9 ⊢ (𝑝 = 𝑧 → (𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
104	103	2rexbidv 3202	. . . . . . . 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 3646	. . . . . . 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 2727	. 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑀‘𝑁) = {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}})
111		dfiunv2 4999	. 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 2782	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 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707   ≠ wne 2925  ∃wrex 3053  {crab 3405  Vcvv 3447   ∖ cdif 3911  {csn 4589  {cpr 4591  ∪ ciun 4955   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387  1c1 11069  2c2 12241  ⟨“cs3 14808  Vtxcvtx 28923  Edgcedg 28974  UMGraphcumgr 29008  USGraphcusgr 29076  FinUSGraphcfusgr 29243   NeighbVtx cnbgr 29259   WSPathsN cwwspthsn 29758   WSPathsNOn cwwspthsnon 29759

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-ac2 10416  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 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-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-er 8671  df-map 8801  df-pm 8802  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-dju 9854  df-card 9892  df-ac 10069  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-xnn0 12516  df-z 12530  df-uz 12794  df-fz 13469  df-fzo 13616  df-hash 14296  df-word 14479  df-concat 14536  df-s1 14561  df-s2 14814  df-s3 14815  df-edg 28975  df-uhgr 28985  df-upgr 29009  df-umgr 29010  df-uspgr 29077  df-usgr 29078  df-fusgr 29244  df-nbgr 29260  df-wlks 29527  df-wlkson 29528  df-trls 29620  df-trlson 29621  df-pths 29644  df-spths 29645  df-pthson 29646  df-spthson 29647  df-wwlks 29760  df-wwlksn 29761  df-wwlksnon 29762  df-wspthsn 29763  df-wspthsnon 29764

This theorem is referenced by:  fusgreghash2wspv  30264