Theorem fusgr2wsp2nb 28107

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 28106	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (𝑧 ∈ (𝑀‘𝑁) ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁)))
4	3	adantl 484	. . . 4 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑧 ∈ (𝑀‘𝑁) ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁)))
5	1	wspthsnwspthsnon 27689	. . . . . . 7 ⊢ (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦))
6		fusgrusgr 27098	. . . . . . . . . 10 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)
7	6	adantr 483	. . . . . . . . 9 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ USGraph)
8		eqid 2821	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
9	1, 8	usgr2wspthon 27738	. . . . . . . . 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 582	. . . . . . . 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 3299	. . . . . . 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	syl5bb 285	. . . . . 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 4576	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)
16	15	bicomi 226	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨“𝑥𝑁𝑦”⟩ ↔ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})
17	16	anbi2i 624	. . . . . . . . . 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 767	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑧‘1) = 𝑁)) → 𝑁 ∈ 𝑉)
20		anass 471	. . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ (𝑥 ≠ 𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))
23		nesym 3072	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑦 = 𝑥)
24		prcom 4661	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑚, 𝑦} = {𝑦, 𝑚}
25	24	eleq1i 2903	. . . . . . . . . . . . . . . . . . 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 624	. . . . . . . . . . . . . . . . 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 277	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≠ 𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
29	28	anbi1i 625	. . . . . . . . . . . . . . 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 299	. . . . . . . . . . . . . 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 4663	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑁 → {𝑥, 𝑚} = {𝑥, 𝑁})
32	31	eleq1d 2897	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑁 → ({𝑥, 𝑚} ∈ (Edg‘𝐺) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
33		preq2 4663	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑁 → {𝑦, 𝑚} = {𝑦, 𝑁})
34	33	eleq1d 2897	. . . . . . . . . . . . . . . . 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 14226	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑁 → ⟨“𝑥𝑚𝑦”⟩ = ⟨“𝑥𝑁𝑦”⟩)
38	37	eqeq2d 2832	. . . . . . . . . . . . . . 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	syl5bb 285	. . . . . . . . . . . . 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 484	. . . . . . . . . . . 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 6663	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = (⟨“𝑥𝑚𝑦”⟩‘1))
43		s3fv1 14248	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ V → (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚)
44	43	elv 3499	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚
45	42, 44	syl6eq 2872	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = 𝑚)
46	45	eqeq1d 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → ((𝑧‘1) = 𝑁 ↔ 𝑚 = 𝑁))
47	46	biimpd 231	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → ((𝑧‘1) = 𝑁 → 𝑚 = 𝑁))
48	47	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥 ≠ 𝑦) → ((𝑧‘1) = 𝑁 → 𝑚 = 𝑁))
49	48	adantr 483	. . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . 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 409	. . . . . . . . . . . 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 3616	. . . . . . . . . . 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 581	. . . . . . . . . 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 644	. . . . . . . . . . 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 26958	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
58	1, 8	umgrpredgv 26919	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → (𝑥 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))
59	58	simpld 497	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → 𝑥 ∈ 𝑉)
60	59	ex 415	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐺 ∈ UMGraph → ({𝑥, 𝑁} ∈ (Edg‘𝐺) → 𝑥 ∈ 𝑉))
61	1, 8	umgrpredgv 26919	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → (𝑦 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))
62	61	simpld 497	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → 𝑦 ∈ 𝑉)
63	62	expcom 416	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑦, 𝑁} ∈ (Edg‘𝐺) → (𝐺 ∈ UMGraph → 𝑦 ∈ 𝑉))
64	63	adantr 483	. . . . . . . . . . . . . . . . . . . 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 610	. . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 14 ⊢ ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
71	70	impcom 410	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
72		fveq1 6663	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨“𝑥𝑁𝑦”⟩ → (𝑧‘1) = (⟨“𝑥𝑁𝑦”⟩‘1))
73	72	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → (𝑧‘1) = (⟨“𝑥𝑁𝑦”⟩‘1))
74		s3fv1 14248	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ 𝑉 → (⟨“𝑥𝑁𝑦”⟩‘1) = 𝑁)
75	74	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (⟨“𝑥𝑁𝑦”⟩‘1) = 𝑁)
76	73, 75	sylan9eqr 2878	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → (𝑧‘1) = 𝑁)
77	71, 76	jca 514	. . . . . . . . . . . 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 415	. . . . . . . . . . 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 565	. . . . . . . . . 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 313	. . . . . . . . 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 27129	. . . . . . . . . . . . 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 483	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
84		eldif 3945	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}))
85	8	nbusgreledg 27129	. . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
88		velsn 4576	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
89	88	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥))
90	89	notbid 320	. . . . . . . . . . . . 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	syl5bb 285	. . . . . . . . . . 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 313	. . . . . . . 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	syl5bbr 287	. . . . . 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 3303	. . . . . . 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 625	. . . . . 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 3303	. . . . . 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 316	. . . . 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 3497	. . . . . . . 8 ⊢ 𝑧 ∈ V
103		eleq1w 2895	. . . . . . . . 9 ⊢ (𝑝 = 𝑧 → (𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
104	103	2rexbidv 3300	. . . . . . . 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 3666	. . . . . . 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 226	. . . . . 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 307	. . . 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 281	. . 3 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑧 ∈ (𝑀‘𝑁) ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}))
110	109	eqrdv 2819	. 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑀‘𝑁) = {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}})
111		dfiunv2 4952	. 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	syl6eqr 2874	1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑀‘𝑁) = ∪ 𝑥 ∈ (𝐺 NeighbVtx 𝑁)∪ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){⟨“𝑥𝑁𝑦”⟩})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  {cab 2799   ≠ wne 3016  ∃wrex 3139  {crab 3142  Vcvv 3494   ∖ cdif 3932  {csn 4560  {cpr 4562  ∪ ciun 4911   ↦ cmpt 5138  ‘cfv 6349  (class class class)co 7150  1c1 10532  2c2 11686  ⟨“cs3 14198  Vtxcvtx 26775  Edgcedg 26826  UMGraphcumgr 26860  USGraphcusgr 26928  FinUSGraphcfusgr 27092   NeighbVtx cnbgr 27108   WSPathsN cwwspthsn 27600   WSPathsNOn cwwspthsnon 27601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-ac2 9879  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-pm 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-dju 9324  df-card 9362  df-ac 9536  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-xnn0 11962  df-z 11976  df-uz 12238  df-fz 12887  df-fzo 13028  df-hash 13685  df-word 13856  df-concat 13917  df-s1 13944  df-s2 14204  df-s3 14205  df-edg 26827  df-uhgr 26837  df-upgr 26861  df-umgr 26862  df-uspgr 26929  df-usgr 26930  df-fusgr 27093  df-nbgr 27109  df-wlks 27375  df-wlkson 27376  df-trls 27468  df-trlson 27469  df-pths 27491  df-spths 27492  df-pthson 27493  df-spthson 27494  df-wwlks 27602  df-wwlksn 27603  df-wwlksnon 27604  df-wspthsn 27605  df-wspthsnon 27606

This theorem is referenced by:  fusgreghash2wspv  28108