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Theorem fusgr2wsp2nb 30131
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 30130 . . . . 5 (𝑁𝑉 → (𝑧 ∈ (𝑀𝑁) ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁)))
43adantl 481 . . . 4 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑧 ∈ (𝑀𝑁) ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁)))
51wspthsnwspthsnon 29714 . . . . . . 7 (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥𝑉𝑦𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦))
6 fusgrusgr 29122 . . . . . . . . . 10 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)
76adantr 480 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → 𝐺 ∈ USGraph)
8 eqid 2727 . . . . . . . . . 10 (Edg‘𝐺) = (Edg‘𝐺)
91, 8usgr2wspthon 29763 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ (𝑥𝑉𝑦𝑉)) → (𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
107, 9sylan 579 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
11102rexbidva 3212 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (∃𝑥𝑉𝑦𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
125, 11bitrid 283 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
1312anbi1d 629 . . . . 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 1947 . . . . . . 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 4640 . . . . . . . . . . . 12 (𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)
1615bicomi 223 . . . . . . . . . . 11 (𝑧 = ⟨“𝑥𝑁𝑦”⟩ ↔ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})
1716anbi2i 622 . . . . . . . . . 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 768 . . . . . . . . . . . 12 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁)) → 𝑁𝑉)
20 anass 468 . . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . . . . 17 ((𝑥𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ (𝑥𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))
23 nesym 2992 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑦 ↔ ¬ 𝑦 = 𝑥)
24 prcom 4732 . . . . . . . . . . . . . . . . . . . 20 {𝑚, 𝑦} = {𝑦, 𝑚}
2524eleq1i 2819 . . . . . . . . . . . . . . . . . . 19 ({𝑚, 𝑦} ∈ (Edg‘𝐺) ↔ {𝑦, 𝑚} ∈ (Edg‘𝐺))
2623, 25anbi12ci 627 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)) ↔ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))
2726anbi2i 622 . . . . . . . . . . . . . . . . 17 (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ (𝑥𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
2822, 27bitri 275 . . . . . . . . . . . . . . . 16 ((𝑥𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
2928anbi1i 623 . . . . . . . . . . . . . . 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 297 . . . . . . . . . . . . . 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 4734 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑁 → {𝑥, 𝑚} = {𝑥, 𝑁})
3231eleq1d 2813 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑁 → ({𝑥, 𝑚} ∈ (Edg‘𝐺) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
33 preq2 4734 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑁 → {𝑦, 𝑚} = {𝑦, 𝑁})
3433eleq1d 2813 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑁 → ({𝑦, 𝑚} ∈ (Edg‘𝐺) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
3534anbi1d 629 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑁 → (({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
3632, 35anbi12d 630 . . . . . . . . . . . . . . 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 14845 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑁 → ⟨“𝑥𝑚𝑦”⟩ = ⟨“𝑥𝑁𝑦”⟩)
3837eqeq2d 2738 . . . . . . . . . . . . . . 15 (𝑚 = 𝑁 → (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ↔ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))
3936, 38anbi12d 630 . . . . . . . . . . . . . 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, 39bitrid 283 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 6890 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = (⟨“𝑥𝑚𝑦”⟩‘1))
43 s3fv1 14867 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ V → (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚)
4443elv 3475 . . . . . . . . . . . . . . . . . . . 20 (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚
4542, 44eqtrdi 2783 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = 𝑚)
4645eqeq1d 2729 . . . . . . . . . . . . . . . . . 18 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → ((𝑧‘1) = 𝑁𝑚 = 𝑁))
4746biimpd 228 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → ((𝑧‘1) = 𝑁𝑚 = 𝑁))
4847adantr 480 . . . . . . . . . . . . . . . 16 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) → ((𝑧‘1) = 𝑁𝑚 = 𝑁))
4948adantr 480 . . . . . . . . . . . . . . 15 (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → ((𝑧‘1) = 𝑁𝑚 = 𝑁))
5049com12 32 . . . . . . . . . . . . . 14 ((𝑧‘1) = 𝑁 → (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → 𝑚 = 𝑁))
5150ad2antll 728 . . . . . . . . . . . . 13 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁)) → (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → 𝑚 = 𝑁))
5251imp 406 . . . . . . . . . . . 12 ((((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁)) ∧ ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) → 𝑚 = 𝑁)
5319, 41, 52rspcebdv 3601 . . . . . . . . . . 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”⟩)))
5453pm5.32da 578 . . . . . . . . . 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”⟩))))
55 an32 645 . . . . . . . . . . 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)))))
5655a1i 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))))))
57 usgrumgr 28981 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
581, 8umgrpredgv 28940 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → (𝑥𝑉𝑁𝑉))
5958simpld 494 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → 𝑥𝑉)
6059ex 412 . . . . . . . . . . . . . . . . . . 19 (𝐺 ∈ UMGraph → ({𝑥, 𝑁} ∈ (Edg‘𝐺) → 𝑥𝑉))
611, 8umgrpredgv 28940 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → (𝑦𝑉𝑁𝑉))
6261simpld 494 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → 𝑦𝑉)
6362expcom 413 . . . . . . . . . . . . . . . . . . . . 21 ({𝑦, 𝑁} ∈ (Edg‘𝐺) → (𝐺 ∈ UMGraph → 𝑦𝑉))
6463adantr 480 . . . . . . . . . . . . . . . . . . . 20 (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) → (𝐺 ∈ UMGraph → 𝑦𝑉))
6564com12 32 . . . . . . . . . . . . . . . . . . 19 (𝐺 ∈ UMGraph → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) → 𝑦𝑉))
6660, 65anim12d 608 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ UMGraph → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → (𝑥𝑉𝑦𝑉)))
676, 57, 663syl 18 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ FinUSGraph → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → (𝑥𝑉𝑦𝑉)))
6867adantr 480 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → (𝑥𝑉𝑦𝑉)))
6968com12 32 . . . . . . . . . . . . . . 15 (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑥𝑉𝑦𝑉)))
7069adantr 480 . . . . . . . . . . . . . 14 ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑥𝑉𝑦𝑉)))
7170impcom 407 . . . . . . . . . . . . 13 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → (𝑥𝑉𝑦𝑉))
72 fveq1 6890 . . . . . . . . . . . . . . 15 (𝑧 = ⟨“𝑥𝑁𝑦”⟩ → (𝑧‘1) = (⟨“𝑥𝑁𝑦”⟩‘1))
7372adantl 481 . . . . . . . . . . . . . 14 ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → (𝑧‘1) = (⟨“𝑥𝑁𝑦”⟩‘1))
74 s3fv1 14867 . . . . . . . . . . . . . . 15 (𝑁𝑉 → (⟨“𝑥𝑁𝑦”⟩‘1) = 𝑁)
7574adantl 481 . . . . . . . . . . . . . 14 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (⟨“𝑥𝑁𝑦”⟩‘1) = 𝑁)
7673, 75sylan9eqr 2789 . . . . . . . . . . . . 13 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → (𝑧‘1) = 𝑁)
7771, 76jca 511 . . . . . . . . . . . 12 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁))
7877ex 412 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁)))
7978pm4.71rd 562 . . . . . . . . . 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”⟩))))
8054, 56, 793bitr4d 311 . . . . . . . . 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”⟩)))
818nbusgreledg 29153 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
826, 81syl 17 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
8382adantr 480 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
84 eldif 3954 . . . . . . . . . . . 12 (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}))
858nbusgreledg 29153 . . . . . . . . . . . . . . 15 (𝐺 ∈ USGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
866, 85syl 17 . . . . . . . . . . . . . 14 (𝐺 ∈ FinUSGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
8786adantr 480 . . . . . . . . . . . . 13 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
88 velsn 4640 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
8988a1i 11 . . . . . . . . . . . . . 14 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥))
9089notbid 318 . . . . . . . . . . . . 13 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (¬ 𝑦 ∈ {𝑥} ↔ ¬ 𝑦 = 𝑥))
9187, 90anbi12d 630 . . . . . . . . . . . 12 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
9284, 91bitrid 283 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
9383, 92anbi12d 630 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ↔ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))))
9493anbi1d 629 . . . . . . . . 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”⟩})))
9518, 80, 943bitr4d 311 . . . . . . . 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”⟩})))
96952exbidv 1920 . . . . . . 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”⟩})))
9714, 96bitr3id 285 . . . . . 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”⟩})))
98 r2ex 3190 . . . . . . 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)))))
9998anbi1i 623 . . . . . 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))
100 r2ex 3190 . . . . . 6 (∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ ∃𝑥𝑦((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
10197, 99, 1003bitr4g 314 . . . . 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”⟩}))
102 vex 3473 . . . . . . . 8 𝑧 ∈ V
103 eleq1w 2811 . . . . . . . . 9 (𝑝 = 𝑧 → (𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
1041032rexbidv 3214 . . . . . . . 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”⟩}))
105102, 104elab 3665 . . . . . . 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”⟩})
106105bicomi 223 . . . . . 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”⟩}})
107106a1i 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”⟩}}))
10813, 101, 1073bitrd 305 . . . 4 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁) ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}))
1094, 108bitrd 279 . . 3 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑧 ∈ (𝑀𝑁) ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}))
110109eqrdv 2725 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑀𝑁) = {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}})
111 dfiunv2 5032 . 2 𝑥 ∈ (𝐺 NeighbVtx 𝑁) 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){⟨“𝑥𝑁𝑦”⟩} = {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}
112110, 111eqtr4di 2785 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 205  wa 395   = wceq 1534  wex 1774  wcel 2099  {cab 2704  wne 2935  wrex 3065  {crab 3427  Vcvv 3469  cdif 3941  {csn 4624  {cpr 4626   ciun 4991  cmpt 5225  cfv 6542  (class class class)co 7414  1c1 11131  2c2 12289  ⟨“cs3 14817  Vtxcvtx 28796  Edgcedg 28847  UMGraphcumgr 28881  USGraphcusgr 28949  FinUSGraphcfusgr 29116   NeighbVtx cnbgr 29132   WSPathsN cwwspthsn 29626   WSPathsNOn cwwspthsnon 29627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-ac2 10478  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ifp 1062  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-map 8838  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9916  df-card 9954  df-ac 10131  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-fz 13509  df-fzo 13652  df-hash 14314  df-word 14489  df-concat 14545  df-s1 14570  df-s2 14823  df-s3 14824  df-edg 28848  df-uhgr 28858  df-upgr 28882  df-umgr 28883  df-uspgr 28950  df-usgr 28951  df-fusgr 29117  df-nbgr 29133  df-wlks 29400  df-wlkson 29401  df-trls 29493  df-trlson 29494  df-pths 29517  df-spths 29518  df-pthson 29519  df-spthson 29520  df-wwlks 29628  df-wwlksn 29629  df-wwlksnon 29630  df-wspthsn 29631  df-wspthsnon 29632
This theorem is referenced by:  fusgreghash2wspv  30132
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