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Theorem fusgr2wsp2nb 27742
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 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
Assertion
Ref Expression
fusgr2wsp2nb ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑀𝑁) = 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.
StepHypRef Expression
1 frgrhash2wsp.v . . . . . 6 𝑉 = (Vtx‘𝐺)
2 fusgreg2wsp.m . . . . . 6 𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
31, 2fusgreg2wsplem 27741 . . . . 5 (𝑁𝑉 → (𝑧 ∈ (𝑀𝑁) ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁)))
43adantl 475 . . . 4 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑧 ∈ (𝑀𝑁) ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁)))
51wspthsnwspthsnon 27296 . . . . . . 7 (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥𝑉𝑦𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦))
6 fusgrusgr 26669 . . . . . . . . . 10 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)
76adantr 474 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → 𝐺 ∈ USGraph)
8 eqid 2778 . . . . . . . . . 10 (Edg‘𝐺) = (Edg‘𝐺)
91, 8usgr2wspthon 27345 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ (𝑥𝑉𝑦𝑉)) → (𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
107, 9sylan 575 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
11102rexbidva 3241 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (∃𝑥𝑉𝑦𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
125, 11syl5bb 275 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
1312anbi1d 623 . . . . 5 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁) ↔ (∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ (𝑧‘1) = 𝑁)))
14 19.41vv 1993 . . . . . . 7 (∃𝑥𝑦(((𝑥𝑉𝑦𝑉) ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ (∃𝑥𝑦((𝑥𝑉𝑦𝑉) ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁))
15 velsn 4414 . . . . . . . . . . . 12 (𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)
1615bicomi 216 . . . . . . . . . . 11 (𝑧 = ⟨“𝑥𝑁𝑦”⟩ ↔ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})
1716anbi2i 616 . . . . . . . . . 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”⟩}))
1817a1i 11 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((({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 759 . . . . . . . . . . . 12 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁)) → 𝑁𝑉)
20 anass 462 . . . . . . . . . . . . . . 15 (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ (𝑥𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
21 ancom 454 . . . . . . . . . . . . . . 15 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ (𝑥𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ↔ ((𝑥𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ 𝑧 = ⟨“𝑥𝑚𝑦”⟩))
22 an12 635 . . . . . . . . . . . . . . . . 17 ((𝑥𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ (𝑥𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))
23 nesym 3025 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑦 ↔ ¬ 𝑦 = 𝑥)
24 prcom 4499 . . . . . . . . . . . . . . . . . . . 20 {𝑚, 𝑦} = {𝑦, 𝑚}
2524eleq1i 2850 . . . . . . . . . . . . . . . . . . 19 ({𝑚, 𝑦} ∈ (Edg‘𝐺) ↔ {𝑦, 𝑚} ∈ (Edg‘𝐺))
2623, 25anbi12ci 621 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)) ↔ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))
2726anbi2i 616 . . . . . . . . . . . . . . . . 17 (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ (𝑥𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
2822, 27bitri 267 . . . . . . . . . . . . . . . 16 ((𝑥𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
2928anbi1i 617 . . . . . . . . . . . . . . 15 (((𝑥𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ 𝑧 = ⟨“𝑥𝑚𝑦”⟩) ↔ (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑚𝑦”⟩))
3020, 21, 293bitri 289 . . . . . . . . . . . . . 14 (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑚𝑦”⟩))
31 preq2 4501 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑁 → {𝑥, 𝑚} = {𝑥, 𝑁})
3231eleq1d 2844 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑁 → ({𝑥, 𝑚} ∈ (Edg‘𝐺) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
33 preq2 4501 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑁 → {𝑦, 𝑚} = {𝑦, 𝑁})
3433eleq1d 2844 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑁 → ({𝑦, 𝑚} ∈ (Edg‘𝐺) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
3534anbi1d 623 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑁 → (({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
3632, 35anbi12d 624 . . . . . . . . . . . . . . 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 14021 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑁 → ⟨“𝑥𝑚𝑦”⟩ = ⟨“𝑥𝑁𝑦”⟩)
3837eqeq2d 2788 . . . . . . . . . . . . . . 15 (𝑚 = 𝑁 → (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ↔ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))
3936, 38anbi12d 624 . . . . . . . . . . . . . 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”⟩)))
4030, 39syl5bb 275 . . . . . . . . . . . . 13 (𝑚 = 𝑁 → (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))
4140adantl 475 . . . . . . . . . . . 12 ((((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁)) ∧ 𝑚 = 𝑁) → (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))
42 fveq1 6445 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = (⟨“𝑥𝑚𝑦”⟩‘1))
43 vex 3401 . . . . . . . . . . . . . . . . . . . . 21 𝑚 ∈ V
44 s3fv1 14043 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ V → (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚)
4543, 44ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚
4642, 45syl6eq 2830 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = 𝑚)
4746eqeq1d 2780 . . . . . . . . . . . . . . . . . 18 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → ((𝑧‘1) = 𝑁𝑚 = 𝑁))
4847biimpd 221 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → ((𝑧‘1) = 𝑁𝑚 = 𝑁))
4948adantr 474 . . . . . . . . . . . . . . . 16 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) → ((𝑧‘1) = 𝑁𝑚 = 𝑁))
5049adantr 474 . . . . . . . . . . . . . . 15 (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → ((𝑧‘1) = 𝑁𝑚 = 𝑁))
5150com12 32 . . . . . . . . . . . . . 14 ((𝑧‘1) = 𝑁 → (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → 𝑚 = 𝑁))
5251ad2antll 719 . . . . . . . . . . . . 13 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁)) → (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → 𝑚 = 𝑁))
5352imp 397 . . . . . . . . . . . 12 ((((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁)) ∧ ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) → 𝑚 = 𝑁)
5419, 41, 53rspcebdv 3516 . . . . . . . . . . 11 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁)) → (∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))
5554pm5.32da 574 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁) ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ↔ (((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))))
56 an32 636 . . . . . . . . . . 11 ((((𝑥𝑉𝑦𝑉) ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ (((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁) ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
5756a1i 11 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((((𝑥𝑉𝑦𝑉) ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ (((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁) ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))))
58 usgrumgr 26528 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
591, 8umgrpredgv 26489 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → (𝑥𝑉𝑁𝑉))
6059simpld 490 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → 𝑥𝑉)
6160ex 403 . . . . . . . . . . . . . . . . . . 19 (𝐺 ∈ UMGraph → ({𝑥, 𝑁} ∈ (Edg‘𝐺) → 𝑥𝑉))
621, 8umgrpredgv 26489 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → (𝑦𝑉𝑁𝑉))
6362simpld 490 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → 𝑦𝑉)
6463expcom 404 . . . . . . . . . . . . . . . . . . . . 21 ({𝑦, 𝑁} ∈ (Edg‘𝐺) → (𝐺 ∈ UMGraph → 𝑦𝑉))
6564adantr 474 . . . . . . . . . . . . . . . . . . . 20 (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) → (𝐺 ∈ UMGraph → 𝑦𝑉))
6665com12 32 . . . . . . . . . . . . . . . . . . 19 (𝐺 ∈ UMGraph → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) → 𝑦𝑉))
6761, 66anim12d 602 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ UMGraph → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → (𝑥𝑉𝑦𝑉)))
686, 58, 673syl 18 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ FinUSGraph → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → (𝑥𝑉𝑦𝑉)))
6968adantr 474 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → (𝑥𝑉𝑦𝑉)))
7069com12 32 . . . . . . . . . . . . . . 15 (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑥𝑉𝑦𝑉)))
7170adantr 474 . . . . . . . . . . . . . 14 ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑥𝑉𝑦𝑉)))
7271impcom 398 . . . . . . . . . . . . 13 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → (𝑥𝑉𝑦𝑉))
73 fveq1 6445 . . . . . . . . . . . . . . 15 (𝑧 = ⟨“𝑥𝑁𝑦”⟩ → (𝑧‘1) = (⟨“𝑥𝑁𝑦”⟩‘1))
7473adantl 475 . . . . . . . . . . . . . 14 ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → (𝑧‘1) = (⟨“𝑥𝑁𝑦”⟩‘1))
75 s3fv1 14043 . . . . . . . . . . . . . . 15 (𝑁𝑉 → (⟨“𝑥𝑁𝑦”⟩‘1) = 𝑁)
7675adantl 475 . . . . . . . . . . . . . 14 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (⟨“𝑥𝑁𝑦”⟩‘1) = 𝑁)
7774, 76sylan9eqr 2836 . . . . . . . . . . . . 13 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → (𝑧‘1) = 𝑁)
7872, 77jca 507 . . . . . . . . . . . 12 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁))
7978ex 403 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁)))
8079pm4.71rd 558 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) ↔ (((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))))
8155, 57, 803bitr4d 303 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((((𝑥𝑉𝑦𝑉) ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))
828nbusgreledg 26700 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
836, 82syl 17 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
8483adantr 474 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
85 eldif 3802 . . . . . . . . . . . 12 (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}))
868nbusgreledg 26700 . . . . . . . . . . . . . . 15 (𝐺 ∈ USGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
876, 86syl 17 . . . . . . . . . . . . . 14 (𝐺 ∈ FinUSGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
8887adantr 474 . . . . . . . . . . . . 13 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
89 velsn 4414 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
9089a1i 11 . . . . . . . . . . . . . 14 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥))
9190notbid 310 . . . . . . . . . . . . 13 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (¬ 𝑦 ∈ {𝑥} ↔ ¬ 𝑦 = 𝑥))
9288, 91anbi12d 624 . . . . . . . . . . . 12 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
9385, 92syl5bb 275 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
9484, 93anbi12d 624 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ↔ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))))
9594anbi1d 623 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}) ↔ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})))
9618, 81, 953bitr4d 303 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((((𝑥𝑉𝑦𝑉) ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ ((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})))
97962exbidv 1967 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (∃𝑥𝑦(((𝑥𝑉𝑦𝑉) ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ ∃𝑥𝑦((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})))
9814, 97syl5bbr 277 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((∃𝑥𝑦((𝑥𝑉𝑦𝑉) ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁) ↔ ∃𝑥𝑦((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})))
99 r2ex 3246 . . . . . . 7 (∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ∃𝑥𝑦((𝑥𝑉𝑦𝑉) ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
10099anbi1i 617 . . . . . 6 ((∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ (𝑧‘1) = 𝑁) ↔ (∃𝑥𝑦((𝑥𝑉𝑦𝑉) ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ∧ (𝑧‘1) = 𝑁))
101 r2ex 3246 . . . . . 6 (∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ ∃𝑥𝑦((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
10298, 100, 1013bitr4g 306 . . . . 5 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ (𝑧‘1) = 𝑁) ↔ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
103 vex 3401 . . . . . . . 8 𝑧 ∈ V
104 eleq1w 2842 . . . . . . . . 9 (𝑝 = 𝑧 → (𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
1051042rexbidv 3242 . . . . . . . 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”⟩}))
106103, 105elab 3558 . . . . . . 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”⟩})
107106bicomi 216 . . . . . 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”⟩}})
108107a1i 11 . . . . 5 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}))
10913, 102, 1083bitrd 297 . . . 4 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁) ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}))
1104, 109bitrd 271 . . 3 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑧 ∈ (𝑀𝑁) ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}))
111110eqrdv 2776 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑀𝑁) = {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}})
112 dfiunv2 4789 . 2 𝑥 ∈ (𝐺 NeighbVtx 𝑁) 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){⟨“𝑥𝑁𝑦”⟩} = {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}
113111, 112syl6eqr 2832 1 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑀𝑁) = 𝑥 ∈ (𝐺 NeighbVtx 𝑁) 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){⟨“𝑥𝑁𝑦”⟩})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386   = wceq 1601  wex 1823  wcel 2107  {cab 2763  wne 2969  wrex 3091  {crab 3094  Vcvv 3398  cdif 3789  {csn 4398  {cpr 4400   ciun 4753  cmpt 4965  cfv 6135  (class class class)co 6922  1c1 10273  2c2 11430  ⟨“cs3 13993  Vtxcvtx 26344  Edgcedg 26395  UMGraphcumgr 26429  USGraphcusgr 26498  FinUSGraphcfusgr 26663   NeighbVtx cnbgr 26679   WSPathsN cwwspthsn 27177   WSPathsNOn cwwspthsnon 27178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-ac2 9620  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ifp 1047  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-ac 9272  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-hash 13436  df-word 13600  df-concat 13661  df-s1 13686  df-s2 13999  df-s3 14000  df-edg 26396  df-uhgr 26406  df-upgr 26430  df-umgr 26431  df-uspgr 26499  df-usgr 26500  df-fusgr 26664  df-nbgr 26680  df-wlks 26947  df-wlkson 26948  df-trls 27043  df-trlson 27044  df-pths 27068  df-spths 27069  df-pthson 27070  df-spthson 27071  df-wwlks 27179  df-wwlksn 27180  df-wwlksnon 27181  df-wspthsn 27182  df-wspthsnon 27183
This theorem is referenced by:  fusgreghash2wspv  27743
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