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Theorem fusgr2wsp2nb 29278
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 29277 . . . . 5 (𝑁𝑉 → (𝑧 ∈ (𝑀𝑁) ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁)))
43adantl 482 . . . 4 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑧 ∈ (𝑀𝑁) ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁)))
51wspthsnwspthsnon 28861 . . . . . . 7 (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥𝑉𝑦𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦))
6 fusgrusgr 28270 . . . . . . . . . 10 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)
76adantr 481 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → 𝐺 ∈ USGraph)
8 eqid 2736 . . . . . . . . . 10 (Edg‘𝐺) = (Edg‘𝐺)
91, 8usgr2wspthon 28910 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ (𝑥𝑉𝑦𝑉)) → (𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
107, 9sylan 580 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
11102rexbidva 3211 . . . . . . 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 282 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
1312anbi1d 630 . . . . 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 1954 . . . . . . 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 4602 . . . . . . . . . . . 12 (𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)
1615bicomi 223 . . . . . . . . . . 11 (𝑧 = ⟨“𝑥𝑁𝑦”⟩ ↔ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})
1716anbi2i 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”⟩}))
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 767 . . . . . . . . . . . 12 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁)) → 𝑁𝑉)
20 anass 469 . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . . . 17 ((𝑥𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ (𝑥𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))
23 nesym 3000 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑦 ↔ ¬ 𝑦 = 𝑥)
24 prcom 4693 . . . . . . . . . . . . . . . . . . . 20 {𝑚, 𝑦} = {𝑦, 𝑚}
2524eleq1i 2828 . . . . . . . . . . . . . . . . . . 19 ({𝑚, 𝑦} ∈ (Edg‘𝐺) ↔ {𝑦, 𝑚} ∈ (Edg‘𝐺))
2623, 25anbi12ci 628 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)) ↔ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))
2726anbi2i 623 . . . . . . . . . . . . . . . . 17 (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ (𝑥𝑦 ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
2822, 27bitri 274 . . . . . . . . . . . . . . . 16 ((𝑥𝑦 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
2928anbi1i 624 . . . . . . . . . . . . . . 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 296 . . . . . . . . . . . . . 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 4695 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑁 → {𝑥, 𝑚} = {𝑥, 𝑁})
3231eleq1d 2822 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑁 → ({𝑥, 𝑚} ∈ (Edg‘𝐺) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
33 preq2 4695 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑁 → {𝑦, 𝑚} = {𝑦, 𝑁})
3433eleq1d 2822 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑁 → ({𝑦, 𝑚} ∈ (Edg‘𝐺) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
3534anbi1d 630 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑁 → (({𝑦, 𝑚} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
3632, 35anbi12d 631 . . . . . . . . . . . . . . 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 14759 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑁 → ⟨“𝑥𝑚𝑦”⟩ = ⟨“𝑥𝑁𝑦”⟩)
3837eqeq2d 2747 . . . . . . . . . . . . . . 15 (𝑚 = 𝑁 → (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ↔ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))
3936, 38anbi12d 631 . . . . . . . . . . . . . 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 282 . . . . . . . . . . . . 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 482 . . . . . . . . . . . 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 6841 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = (⟨“𝑥𝑚𝑦”⟩‘1))
43 s3fv1 14781 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ V → (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚)
4443elv 3451 . . . . . . . . . . . . . . . . . . . 20 (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚
4542, 44eqtrdi 2792 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = 𝑚)
4645eqeq1d 2738 . . . . . . . . . . . . . . . . . 18 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → ((𝑧‘1) = 𝑁𝑚 = 𝑁))
4746biimpd 228 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → ((𝑧‘1) = 𝑁𝑚 = 𝑁))
4847adantr 481 . . . . . . . . . . . . . . . 16 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) → ((𝑧‘1) = 𝑁𝑚 = 𝑁))
4948adantr 481 . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . 13 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁)) → (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → 𝑚 = 𝑁))
5251imp 407 . . . . . . . . . . . 12 ((((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁)) ∧ ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) → 𝑚 = 𝑁)
5319, 41, 52rspcebdv 3575 . . . . . . . . . . 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 579 . . . . . . . . . 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 644 . . . . . . . . . . 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 28130 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph)
581, 8umgrpredgv 28091 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → (𝑥𝑉𝑁𝑉))
5958simpld 495 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → 𝑥𝑉)
6059ex 413 . . . . . . . . . . . . . . . . . . 19 (𝐺 ∈ UMGraph → ({𝑥, 𝑁} ∈ (Edg‘𝐺) → 𝑥𝑉))
611, 8umgrpredgv 28091 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → (𝑦𝑉𝑁𝑉))
6261simpld 495 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → 𝑦𝑉)
6362expcom 414 . . . . . . . . . . . . . . . . . . . . 21 ({𝑦, 𝑁} ∈ (Edg‘𝐺) → (𝐺 ∈ UMGraph → 𝑦𝑉))
6463adantr 481 . . . . . . . . . . . . . . . . . . . 20 (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) → (𝐺 ∈ UMGraph → 𝑦𝑉))
6564com12 32 . . . . . . . . . . . . . . . . . . 19 (𝐺 ∈ UMGraph → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) → 𝑦𝑉))
6660, 65anim12d 609 . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑥𝑉𝑦𝑉)))
7170impcom 408 . . . . . . . . . . . . 13 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → (𝑥𝑉𝑦𝑉))
72 fveq1 6841 . . . . . . . . . . . . . . 15 (𝑧 = ⟨“𝑥𝑁𝑦”⟩ → (𝑧‘1) = (⟨“𝑥𝑁𝑦”⟩‘1))
7372adantl 482 . . . . . . . . . . . . . 14 ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → (𝑧‘1) = (⟨“𝑥𝑁𝑦”⟩‘1))
74 s3fv1 14781 . . . . . . . . . . . . . . 15 (𝑁𝑉 → (⟨“𝑥𝑁𝑦”⟩‘1) = 𝑁)
7574adantl 482 . . . . . . . . . . . . . 14 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (⟨“𝑥𝑁𝑦”⟩‘1) = 𝑁)
7673, 75sylan9eqr 2798 . . . . . . . . . . . . 13 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → (𝑧‘1) = 𝑁)
7771, 76jca 512 . . . . . . . . . . . 12 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁))
7877ex 413 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → ((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁)))
7978pm4.71rd 563 . . . . . . . . . 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 310 . . . . . . . . 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 28301 . . . . . . . . . . . . 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 481 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
84 eldif 3920 . . . . . . . . . . . 12 (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}))
858nbusgreledg 28301 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 13 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
88 velsn 4602 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
8988a1i 11 . . . . . . . . . . . . . 14 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥))
9089notbid 317 . . . . . . . . . . . . 13 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (¬ 𝑦 ∈ {𝑥} ↔ ¬ 𝑦 = 𝑥))
9187, 90anbi12d 631 . . . . . . . . . . . 12 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
9284, 91bitrid 282 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
9383, 92anbi12d 631 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})) ↔ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))))
9493anbi1d 630 . . . . . . . . 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 310 . . . . . . . 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 1927 . . . . . . 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 284 . . . . . 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 3192 . . . . . . 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 624 . . . . . 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 3192 . . . . . 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 313 . . . . 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 3449 . . . . . . . 8 𝑧 ∈ V
103 eleq1w 2820 . . . . . . . . 9 (𝑝 = 𝑧 → (𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
1041032rexbidv 3213 . . . . . . . 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 3630 . . . . . . 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 304 . . . 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 278 . . 3 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑧 ∈ (𝑀𝑁) ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}))
110109eqrdv 2734 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑀𝑁) = {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}})
111 dfiunv2 4995 . 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 2794 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 396   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wne 2943  wrex 3073  {crab 3407  Vcvv 3445  cdif 3907  {csn 4586  {cpr 4588   ciun 4954  cmpt 5188  cfv 6496  (class class class)co 7357  1c1 11052  2c2 12208  ⟨“cs3 14731  Vtxcvtx 27947  Edgcedg 27998  UMGraphcumgr 28032  USGraphcusgr 28100  FinUSGraphcfusgr 28264   NeighbVtx cnbgr 28280   WSPathsN cwwspthsn 28773   WSPathsNOn cwwspthsnon 28774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-s2 14737  df-s3 14738  df-edg 27999  df-uhgr 28009  df-upgr 28033  df-umgr 28034  df-uspgr 28101  df-usgr 28102  df-fusgr 28265  df-nbgr 28281  df-wlks 28547  df-wlkson 28548  df-trls 28640  df-trlson 28641  df-pths 28664  df-spths 28665  df-pthson 28666  df-spthson 28667  df-wwlks 28775  df-wwlksn 28776  df-wwlksnon 28777  df-wspthsn 28778  df-wspthsnon 28779
This theorem is referenced by:  fusgreghash2wspv  29279
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