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Theorem usg2spot2nb 26592
 Description: The set of paths of length 2 with a given vertex in the middle for a finite 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.)
Hypothesis
Ref Expression
usgreghash2spot.m 𝑀 = (𝑎𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎)})
Assertion
Ref Expression
usg2spot2nb ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑀𝑁) = 𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) 𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥}){⟨𝑥, 𝑁, 𝑦⟩})
Distinct variable groups:   𝑡,𝐸,𝑥,𝑦   𝑁,𝑎,𝑡,𝑥,𝑦   𝑉,𝑎,𝑡,𝑥,𝑦   𝐸,𝑎
Allowed substitution hints:   𝑀(𝑥,𝑦,𝑡,𝑎)

Proof of Theorem usg2spot2nb
Dummy variables 𝑚 𝑝 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1056 . . . . 5 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → 𝑁𝑉)
2 3xpexg 6859 . . . . . . 7 (𝑉 ∈ Fin → ((𝑉 × 𝑉) × 𝑉) ∈ V)
3 rabexg 4739 . . . . . . 7 (((𝑉 × 𝑉) × 𝑉) ∈ V → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑁)} ∈ V)
42, 3syl 17 . . . . . 6 (𝑉 ∈ Fin → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑁)} ∈ V)
543ad2ant2 1076 . . . . 5 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑁)} ∈ V)
6 eqeq2 2621 . . . . . . . . 9 (𝑎 = 𝑁 → ((2nd ‘(1st𝑡)) = 𝑎 ↔ (2nd ‘(1st𝑡)) = 𝑁))
76anbi2d 736 . . . . . . . 8 (𝑎 = 𝑁 → ((𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎) ↔ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑁)))
87rabbidv 3164 . . . . . . 7 (𝑎 = 𝑁 → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎)} = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑁)})
9 usgreghash2spot.m . . . . . . 7 𝑀 = (𝑎𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎)})
108, 9fvmptg 6189 . . . . . 6 ((𝑁𝑉 ∧ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑁)} ∈ V) → (𝑀𝑁) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑁)})
1110eleq2d 2673 . . . . 5 ((𝑁𝑉 ∧ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑁)} ∈ V) → (𝑧 ∈ (𝑀𝑁) ↔ 𝑧 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑁)}))
121, 5, 11syl2anc 691 . . . 4 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑧 ∈ (𝑀𝑁) ↔ 𝑧 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑁)}))
13 eleq1 2676 . . . . . . . 8 (𝑡 = 𝑧 → (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ↔ 𝑧 ∈ (𝑉 2SPathsOt 𝐸)))
14 fveq2 6103 . . . . . . . . . 10 (𝑡 = 𝑧 → (1st𝑡) = (1st𝑧))
1514fveq2d 6107 . . . . . . . . 9 (𝑡 = 𝑧 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑧)))
1615eqeq1d 2612 . . . . . . . 8 (𝑡 = 𝑧 → ((2nd ‘(1st𝑡)) = 𝑁 ↔ (2nd ‘(1st𝑧)) = 𝑁))
1713, 16anbi12d 743 . . . . . . 7 (𝑡 = 𝑧 → ((𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑁) ↔ (𝑧 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑧)) = 𝑁)))
1817elrab 3331 . . . . . 6 (𝑧 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑁)} ↔ (𝑧 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑧 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑧)) = 𝑁)))
1918a1i 11 . . . . 5 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑧 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑁)} ↔ (𝑧 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑧 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑧)) = 𝑁))))
20 usgrav 25867 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
21 el2spthsoton 26406 . . . . . . . . . 10 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑧 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑥𝑉𝑦𝑉 𝑧 ∈ (𝑥(𝑉 2SPathOnOt 𝐸)𝑦)))
2220, 21syl 17 . . . . . . . . 9 (𝑉 USGrph 𝐸 → (𝑧 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑥𝑉𝑦𝑉 𝑧 ∈ (𝑥(𝑉 2SPathOnOt 𝐸)𝑦)))
23223ad2ant1 1075 . . . . . . . 8 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑧 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑥𝑉𝑦𝑉 𝑧 ∈ (𝑥(𝑉 2SPathOnOt 𝐸)𝑦)))
24 usg2spthonot1 26417 . . . . . . . . . 10 ((𝑉 USGrph 𝐸 ∧ (𝑥𝑉𝑦𝑉)) → (𝑧 ∈ (𝑥(𝑉 2SPathOnOt 𝐸)𝑦) ↔ ∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸))))
25243ad2antl1 1216 . . . . . . . . 9 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑧 ∈ (𝑥(𝑉 2SPathOnOt 𝐸)𝑦) ↔ ∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸))))
26252rexbidva 3038 . . . . . . . 8 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (∃𝑥𝑉𝑦𝑉 𝑧 ∈ (𝑥(𝑉 2SPathOnOt 𝐸)𝑦) ↔ ∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸))))
2723, 26bitrd 267 . . . . . . 7 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑧 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸))))
2827anbi1d 737 . . . . . 6 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((𝑧 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑧)) = 𝑁) ↔ (∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁)))
2928anbi2d 736 . . . . 5 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((𝑧 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑧 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑧)) = 𝑁)) ↔ (𝑧 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁))))
30 r19.41vv 3072 . . . . . . 7 (∃𝑥𝑉𝑦𝑉 ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) ↔ (∃𝑥𝑉𝑦𝑉 (∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))
31 ancom 465 . . . . . . 7 ((∃𝑥𝑉𝑦𝑉 (∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) ↔ (𝑧 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑥𝑉𝑦𝑉 (∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁)))
32 r19.41vv 3072 . . . . . . . 8 (∃𝑥𝑉𝑦𝑉 (∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ↔ (∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁))
3332anbi2i 726 . . . . . . 7 ((𝑧 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑥𝑉𝑦𝑉 (∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁)) ↔ (𝑧 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁)))
3430, 31, 333bitrri 286 . . . . . 6 ((𝑧 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁)) ↔ ∃𝑥𝑉𝑦𝑉 ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))
35 velsn 4141 . . . . . . . . . . . . 13 (𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩} ↔ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)
3635bicomi 213 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ↔ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩})
3736anbi2i 726 . . . . . . . . . . 11 (((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) ↔ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}))
3837anbi2i 726 . . . . . . . . . 10 ((((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)) ↔ (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩})))
3938a1i 11 . . . . . . . . 9 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)) ↔ (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}))))
40 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ → (1st𝑧) = (1st ‘⟨𝑥, 𝑚, 𝑦⟩))
4140fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ → (2nd ‘(1st𝑧)) = (2nd ‘(1st ‘⟨𝑥, 𝑚, 𝑦⟩)))
42 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑥 ∈ V
43 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑚 ∈ V
44 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑦 ∈ V
45 ot2ndg 7074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑥 ∈ V ∧ 𝑚 ∈ V ∧ 𝑦 ∈ V) → (2nd ‘(1st ‘⟨𝑥, 𝑚, 𝑦⟩)) = 𝑚)
4642, 43, 44, 45mp3an 1416 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (2nd ‘(1st ‘⟨𝑥, 𝑚, 𝑦⟩)) = 𝑚
4741, 46syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ → (2nd ‘(1st𝑧)) = 𝑚)
4847eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ → ((2nd ‘(1st𝑧)) = 𝑁𝑚 = 𝑁))
49 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((((𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸)) ∧ 𝑥𝑦) ∧ 𝑁𝑉) ∧ (𝑦𝑉𝑥𝑉)) ∧ 𝑁𝑉) → 𝑁𝑉)
50 simplrl 796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((((𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸)) ∧ 𝑥𝑦) ∧ 𝑁𝑉) ∧ (𝑦𝑉𝑥𝑉)) ∧ 𝑁𝑉) → 𝑦𝑉)
51 simplrr 797 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸)) ∧ 𝑥𝑦) → {𝑁, 𝑦} ∈ ran 𝐸)
5251ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((((𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸)) ∧ 𝑥𝑦) ∧ 𝑁𝑉) ∧ (𝑦𝑉𝑥𝑉)) ∧ 𝑁𝑉) → {𝑁, 𝑦} ∈ ran 𝐸)
5349, 50, 523jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((((𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸)) ∧ 𝑥𝑦) ∧ 𝑁𝑉) ∧ (𝑦𝑉𝑥𝑉)) ∧ 𝑁𝑉) → (𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸))
54 nesym 2838 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑥𝑦 ↔ ¬ 𝑦 = 𝑥)
5554biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑥𝑦 → ¬ 𝑦 = 𝑥)
5655ad4antlr 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((((𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸)) ∧ 𝑥𝑦) ∧ 𝑁𝑉) ∧ (𝑦𝑉𝑥𝑉)) ∧ 𝑁𝑉) → ¬ 𝑦 = 𝑥)
5753, 56jca 553 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((((𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸)) ∧ 𝑥𝑦) ∧ 𝑁𝑉) ∧ (𝑦𝑉𝑥𝑉)) ∧ 𝑁𝑉) → ((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥))
58 simplrr 797 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((((𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸)) ∧ 𝑥𝑦) ∧ 𝑁𝑉) ∧ (𝑦𝑉𝑥𝑉)) ∧ 𝑁𝑉) → 𝑥𝑉)
59 prcom 4211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 {𝑥, 𝑁} = {𝑁, 𝑥}
6059eleq1i 2679 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ({𝑥, 𝑁} ∈ ran 𝐸 ↔ {𝑁, 𝑥} ∈ ran 𝐸)
6160biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ({𝑥, 𝑁} ∈ ran 𝐸 → {𝑁, 𝑥} ∈ ran 𝐸)
6261adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸) → {𝑁, 𝑥} ∈ ran 𝐸)
6362ad5antlr 767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((((𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸)) ∧ 𝑥𝑦) ∧ 𝑁𝑉) ∧ (𝑦𝑉𝑥𝑉)) ∧ 𝑁𝑉) → {𝑁, 𝑥} ∈ ran 𝐸)
6449, 58, 633jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((((𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸)) ∧ 𝑥𝑦) ∧ 𝑁𝑉) ∧ (𝑦𝑉𝑥𝑉)) ∧ 𝑁𝑉) → (𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸))
65 simp-5l 804 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((((𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸)) ∧ 𝑥𝑦) ∧ 𝑁𝑉) ∧ (𝑦𝑉𝑥𝑉)) ∧ 𝑁𝑉) → 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)
6657, 64, 65jca32 556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((((𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸)) ∧ 𝑥𝑦) ∧ 𝑁𝑉) ∧ (𝑦𝑉𝑥𝑉)) ∧ 𝑁𝑉) → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))
6766exp31 628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸)) ∧ 𝑥𝑦) ∧ 𝑁𝑉) → ((𝑦𝑉𝑥𝑉) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))
6867exp41 636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ → (({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸) → (𝑥𝑦 → (𝑁𝑉 → ((𝑦𝑉𝑥𝑉) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))))))))
6968a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 = 𝑁 → (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ → (({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸) → (𝑥𝑦 → (𝑁𝑉 → ((𝑦𝑉𝑥𝑉) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))))))
70 oteq2 4350 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 = 𝑁 → ⟨𝑥, 𝑚, 𝑦⟩ = ⟨𝑥, 𝑁, 𝑦⟩)
7170eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 = 𝑁 → (𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))
72 preq2 4213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑚 = 𝑁 → {𝑥, 𝑚} = {𝑥, 𝑁})
7372eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑚 = 𝑁 → ({𝑥, 𝑚} ∈ ran 𝐸 ↔ {𝑥, 𝑁} ∈ ran 𝐸))
74 preq1 4212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑚 = 𝑁 → {𝑚, 𝑦} = {𝑁, 𝑦})
7574eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑚 = 𝑁 → ({𝑚, 𝑦} ∈ ran 𝐸 ↔ {𝑁, 𝑦} ∈ ran 𝐸))
7673, 75anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 = 𝑁 → (({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸) ↔ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸)))
77 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑚 = 𝑁 → (𝑚𝑉𝑁𝑉))
7877imbi1d 330 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑚 = 𝑁 → ((𝑚𝑉 → ((𝑦𝑉𝑥𝑉) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))))) ↔ (𝑁𝑉 → ((𝑦𝑉𝑥𝑉) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))))
7978imbi2d 329 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 = 𝑁 → ((𝑥𝑦 → (𝑚𝑉 → ((𝑦𝑉𝑥𝑉) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))) ↔ (𝑥𝑦 → (𝑁𝑉 → ((𝑦𝑉𝑥𝑉) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))))))))
8076, 79imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 = 𝑁 → ((({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸) → (𝑥𝑦 → (𝑚𝑉 → ((𝑦𝑉𝑥𝑉) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))))))) ↔ (({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸) → (𝑥𝑦 → (𝑁𝑉 → ((𝑦𝑉𝑥𝑉) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))))))
8169, 71, 803imtr4d 282 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑚 = 𝑁 → (𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ → (({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸) → (𝑥𝑦 → (𝑚𝑉 → ((𝑦𝑉𝑥𝑉) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))))))
8281com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ → (𝑚 = 𝑁 → (({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸) → (𝑥𝑦 → (𝑚𝑉 → ((𝑦𝑉𝑥𝑉) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))))))
8348, 82sylbid 229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ → ((2nd ‘(1st𝑧)) = 𝑁 → (({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸) → (𝑥𝑦 → (𝑚𝑉 → ((𝑦𝑉𝑥𝑉) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))))))
8483com24 93 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ → (𝑥𝑦 → (({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸) → ((2nd ‘(1st𝑧)) = 𝑁 → (𝑚𝑉 → ((𝑦𝑉𝑥𝑉) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))))))
8584imp31 447 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) → ((2nd ‘(1st𝑧)) = 𝑁 → (𝑚𝑉 → ((𝑦𝑉𝑥𝑉) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))))
8685com14 94 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦𝑉𝑥𝑉) → ((2nd ‘(1st𝑧)) = 𝑁 → (𝑚𝑉 → (((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))))
8786adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑦𝑉𝑥𝑉) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((2nd ‘(1st𝑧)) = 𝑁 → (𝑚𝑉 → (((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))))
8887imp41 617 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑦𝑉𝑥𝑉) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑚𝑉) ∧ ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸))) → (𝑁𝑉 → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))))
8988com12 32 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁𝑉 → ((((((𝑦𝑉𝑥𝑉) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑚𝑉) ∧ ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸))) → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))))
90893ad2ant3 1077 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((((((𝑦𝑉𝑥𝑉) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑚𝑉) ∧ ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸))) → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))))
9190expdcom 454 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑦𝑉𝑥𝑉) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑚𝑉) → (((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) → ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))
9291rexlimdva 3013 . . . . . . . . . . . . . . . . . . . 20 ((((𝑦𝑉𝑥𝑉) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ (2nd ‘(1st𝑧)) = 𝑁) → (∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) → ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))
9392ex 449 . . . . . . . . . . . . . . . . . . 19 (((𝑦𝑉𝑥𝑉) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((2nd ‘(1st𝑧)) = 𝑁 → (∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) → ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))))))
9493com13 86 . . . . . . . . . . . . . . . . . 18 (∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) → ((2nd ‘(1st𝑧)) = 𝑁 → (((𝑦𝑉𝑥𝑉) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))))))
9594imp 444 . . . . . . . . . . . . . . . . 17 ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) → (((𝑦𝑉𝑥𝑉) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))
9695expcomd 453 . . . . . . . . . . . . . . . 16 ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) → (𝑧 ∈ ((𝑉 × 𝑉) × 𝑉) → ((𝑦𝑉𝑥𝑉) → ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))))))
9796imp 444 . . . . . . . . . . . . . . 15 (((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((𝑦𝑉𝑥𝑉) → ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))
9897expdcom 454 . . . . . . . . . . . . . 14 (𝑦𝑉 → (𝑥𝑉 → (((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) → ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))))))
9998com23 84 . . . . . . . . . . . . 13 (𝑦𝑉 → (((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) → (𝑥𝑉 → ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))))))
10099imp 444 . . . . . . . . . . . 12 ((𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉))) → (𝑥𝑉 → ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)))))
101100impcom 445 . . . . . . . . . . 11 ((𝑥𝑉 ∧ (𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))) → ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))))
102101com12 32 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((𝑥𝑉 ∧ (𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))) → (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))))
103 simprl2 1100 . . . . . . . . . . 11 ((((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)) → 𝑥𝑉)
104 simpll2 1094 . . . . . . . . . . 11 ((((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)) → 𝑦𝑉)
105 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁𝑉𝑁𝑉)
106 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((({𝑁, 𝑦} ∈ ran 𝐸 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)
10754bicomi 213 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑦 = 𝑥𝑥𝑦)
108107biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑦 = 𝑥𝑥𝑦)
109106, 108anim12i 588 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((({𝑁, 𝑦} ∈ ran 𝐸 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) ∧ ¬ 𝑦 = 𝑥) → (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ 𝑥𝑦))
110 prcom 4211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 {𝑁, 𝑥} = {𝑥, 𝑁}
111110eleq1i 2679 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ({𝑁, 𝑥} ∈ ran 𝐸 ↔ {𝑥, 𝑁} ∈ ran 𝐸)
112111biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ({𝑁, 𝑥} ∈ ran 𝐸 → {𝑥, 𝑁} ∈ ran 𝐸)
113112anim1i 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (({𝑁, 𝑥} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸) → ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸))
114113ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (({𝑁, 𝑦} ∈ ran 𝐸 ∧ {𝑁, 𝑥} ∈ ran 𝐸) → ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸))
115114adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((({𝑁, 𝑦} ∈ ran 𝐸 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸))
116115adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((({𝑁, 𝑦} ∈ ran 𝐸 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) ∧ ¬ 𝑦 = 𝑥) → ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸))
117109, 116jca 553 . . . . . . . . . . . . . . . . . . . . . . . 24 (((({𝑁, 𝑦} ∈ ran 𝐸 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) ∧ ¬ 𝑦 = 𝑥) → ((𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸)))
11871anbi1d 737 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑚 = 𝑁 → ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ↔ (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ 𝑥𝑦)))
119118, 76anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = 𝑁 → (((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ↔ ((𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑁} ∈ ran 𝐸 ∧ {𝑁, 𝑦} ∈ ran 𝐸))))
120117, 119syl5ibr 235 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑁 → (((({𝑁, 𝑦} ∈ ran 𝐸 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) ∧ ¬ 𝑦 = 𝑥) → ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸))))
121120adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁𝑉𝑚 = 𝑁) → (((({𝑁, 𝑦} ∈ ran 𝐸 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) ∧ ¬ 𝑦 = 𝑥) → ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸))))
122105, 121rspcimedv 3284 . . . . . . . . . . . . . . . . . . . . 21 (𝑁𝑉 → (((({𝑁, 𝑦} ∈ ran 𝐸 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) ∧ ¬ 𝑦 = 𝑥) → ∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸))))
123122expdcom 454 . . . . . . . . . . . . . . . . . . . 20 ((({𝑁, 𝑦} ∈ ran 𝐸 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → (¬ 𝑦 = 𝑥 → (𝑁𝑉 → ∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)))))
124123exp31 628 . . . . . . . . . . . . . . . . . . 19 ({𝑁, 𝑦} ∈ ran 𝐸 → ({𝑁, 𝑥} ∈ ran 𝐸 → (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ → (¬ 𝑦 = 𝑥 → (𝑁𝑉 → ∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)))))))
125124com15 99 . . . . . . . . . . . . . . . . . 18 (𝑁𝑉 → ({𝑁, 𝑥} ∈ ran 𝐸 → (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ → (¬ 𝑦 = 𝑥 → ({𝑁, 𝑦} ∈ ran 𝐸 → ∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)))))))
126125imp 444 . . . . . . . . . . . . . . . . 17 ((𝑁𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) → (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ → (¬ 𝑦 = 𝑥 → ({𝑁, 𝑦} ∈ ran 𝐸 → ∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸))))))
1271263adant2 1073 . . . . . . . . . . . . . . . 16 ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) → (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ → (¬ 𝑦 = 𝑥 → ({𝑁, 𝑦} ∈ ran 𝐸 → ∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸))))))
128127imp 444 . . . . . . . . . . . . . . 15 (((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → (¬ 𝑦 = 𝑥 → ({𝑁, 𝑦} ∈ ran 𝐸 → ∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)))))
129128com13 86 . . . . . . . . . . . . . 14 ({𝑁, 𝑦} ∈ ran 𝐸 → (¬ 𝑦 = 𝑥 → (((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → ∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)))))
1301293ad2ant3 1077 . . . . . . . . . . . . 13 ((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) → (¬ 𝑦 = 𝑥 → (((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → ∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)))))
131130imp31 447 . . . . . . . . . . . 12 ((((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)) → ∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)))
132 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ → (1st𝑧) = (1st ‘⟨𝑥, 𝑁, 𝑦⟩))
133132fveq2d 6107 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ → (2nd ‘(1st𝑧)) = (2nd ‘(1st ‘⟨𝑥, 𝑁, 𝑦⟩)))
134 simpl 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥𝑉 ∧ (𝑁𝑉𝑦𝑉)) → 𝑥𝑉)
135 simprl 790 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥𝑉 ∧ (𝑁𝑉𝑦𝑉)) → 𝑁𝑉)
136 simprr 792 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥𝑉 ∧ (𝑁𝑉𝑦𝑉)) → 𝑦𝑉)
137134, 135, 1363jca 1235 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑉 ∧ (𝑁𝑉𝑦𝑉)) → (𝑥𝑉𝑁𝑉𝑦𝑉))
138 ot2ndg 7074 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑉𝑁𝑉𝑦𝑉) → (2nd ‘(1st ‘⟨𝑥, 𝑁, 𝑦⟩)) = 𝑁)
139137, 138syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝑉 ∧ (𝑁𝑉𝑦𝑉)) → (2nd ‘(1st ‘⟨𝑥, 𝑁, 𝑦⟩)) = 𝑁)
140133, 139sylan9eqr 2666 . . . . . . . . . . . . . . . . . . . 20 (((𝑥𝑉 ∧ (𝑁𝑉𝑦𝑉)) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → (2nd ‘(1st𝑧)) = 𝑁)
141140exp31 628 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑉 → ((𝑁𝑉𝑦𝑉) → (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ → (2nd ‘(1st𝑧)) = 𝑁)))
142141com23 84 . . . . . . . . . . . . . . . . . 18 (𝑥𝑉 → (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ → ((𝑁𝑉𝑦𝑉) → (2nd ‘(1st𝑧)) = 𝑁)))
1431423ad2ant2 1076 . . . . . . . . . . . . . . . . 17 ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) → (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ → ((𝑁𝑉𝑦𝑉) → (2nd ‘(1st𝑧)) = 𝑁)))
144143imp 444 . . . . . . . . . . . . . . . 16 (((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → ((𝑁𝑉𝑦𝑉) → (2nd ‘(1st𝑧)) = 𝑁))
145144com12 32 . . . . . . . . . . . . . . 15 ((𝑁𝑉𝑦𝑉) → (((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → (2nd ‘(1st𝑧)) = 𝑁))
1461453adant3 1074 . . . . . . . . . . . . . 14 ((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) → (((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → (2nd ‘(1st𝑧)) = 𝑁))
147146adantr 480 . . . . . . . . . . . . 13 (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) → (((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → (2nd ‘(1st𝑧)) = 𝑁))
148147imp 444 . . . . . . . . . . . 12 ((((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)) → (2nd ‘(1st𝑧)) = 𝑁)
149 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥𝑉 ∧ (𝑁𝑉𝑦𝑉)) → ⟨𝑥, 𝑁, 𝑦⟩ = ⟨𝑥, 𝑁, 𝑦⟩)
150 otel3xp 5077 . . . . . . . . . . . . . . . . . . . . . . 23 ((⟨𝑥, 𝑁, 𝑦⟩ = ⟨𝑥, 𝑁, 𝑦⟩ ∧ (𝑥𝑉𝑁𝑉𝑦𝑉)) → ⟨𝑥, 𝑁, 𝑦⟩ ∈ ((𝑉 × 𝑉) × 𝑉))
151149, 137, 150syl2anc 691 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑉 ∧ (𝑁𝑉𝑦𝑉)) → ⟨𝑥, 𝑁, 𝑦⟩ ∈ ((𝑉 × 𝑉) × 𝑉))
152151adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥𝑉 ∧ (𝑁𝑉𝑦𝑉)) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → ⟨𝑥, 𝑁, 𝑦⟩ ∈ ((𝑉 × 𝑉) × 𝑉))
153 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ → (𝑧 ∈ ((𝑉 × 𝑉) × 𝑉) ↔ ⟨𝑥, 𝑁, 𝑦⟩ ∈ ((𝑉 × 𝑉) × 𝑉)))
154153adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥𝑉 ∧ (𝑁𝑉𝑦𝑉)) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → (𝑧 ∈ ((𝑉 × 𝑉) × 𝑉) ↔ ⟨𝑥, 𝑁, 𝑦⟩ ∈ ((𝑉 × 𝑉) × 𝑉)))
155152, 154mpbird 246 . . . . . . . . . . . . . . . . . . . 20 (((𝑥𝑉 ∧ (𝑁𝑉𝑦𝑉)) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉))
156155exp31 628 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑉 → ((𝑁𝑉𝑦𝑉) → (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ → 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉))))
157156com23 84 . . . . . . . . . . . . . . . . . 18 (𝑥𝑉 → (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ → ((𝑁𝑉𝑦𝑉) → 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉))))
1581573ad2ant2 1076 . . . . . . . . . . . . . . . . 17 ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) → (𝑧 = ⟨𝑥, 𝑁, 𝑦⟩ → ((𝑁𝑉𝑦𝑉) → 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉))))
159158imp 444 . . . . . . . . . . . . . . . 16 (((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → ((𝑁𝑉𝑦𝑉) → 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))
160159com12 32 . . . . . . . . . . . . . . 15 ((𝑁𝑉𝑦𝑉) → (((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))
1611603adant3 1074 . . . . . . . . . . . . . 14 ((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) → (((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))
162161adantr 480 . . . . . . . . . . . . 13 (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) → (((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩) → 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))
163162imp 444 . . . . . . . . . . . 12 ((((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)) → 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉))
164131, 148, 163jca31 555 . . . . . . . . . . 11 ((((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)) → ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))
165103, 104, 164jca32 556 . . . . . . . . . 10 ((((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩)) → (𝑥𝑉 ∧ (𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))))
166102, 165impbid1 214 . . . . . . . . 9 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((𝑥𝑉 ∧ (𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))) ↔ (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 = ⟨𝑥, 𝑁, 𝑦⟩))))
167 eldif 3550 . . . . . . . . . . 11 (𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥}) ↔ (𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}))
168 nbgrael 25955 . . . . . . . . . . . . . 14 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ↔ (𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸)))
16920, 168syl 17 . . . . . . . . . . . . 13 (𝑉 USGrph 𝐸 → (𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ↔ (𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸)))
1701693ad2ant1 1075 . . . . . . . . . . . 12 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ↔ (𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸)))
171 velsn 4141 . . . . . . . . . . . . . 14 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
172171a1i 11 . . . . . . . . . . . . 13 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥))
173172notbid 307 . . . . . . . . . . . 12 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (¬ 𝑦 ∈ {𝑥} ↔ ¬ 𝑦 = 𝑥))
174170, 173anbi12d 743 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}) ↔ ((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥)))
175167, 174syl5bb 271 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥}) ↔ ((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥)))
176 nbgrael 25955 . . . . . . . . . . . . 13 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ↔ (𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸)))
17720, 176syl 17 . . . . . . . . . . . 12 (𝑉 USGrph 𝐸 → (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ↔ (𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸)))
1781773ad2ant1 1075 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ↔ (𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸)))
179178anbi1d 737 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}) ↔ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩})))
180175, 179anbi12d 743 . . . . . . . . 9 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥}) ∧ (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩})) ↔ (((𝑁𝑉𝑦𝑉 ∧ {𝑁, 𝑦} ∈ ran 𝐸) ∧ ¬ 𝑦 = 𝑥) ∧ ((𝑁𝑉𝑥𝑉 ∧ {𝑁, 𝑥} ∈ ran 𝐸) ∧ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}))))
18139, 166, 1803bitr4d 299 . . . . . . . 8 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((𝑥𝑉 ∧ (𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))) ↔ (𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥}) ∧ (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}))))
1821812exbidv 1839 . . . . . . 7 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (∃𝑥𝑦(𝑥𝑉 ∧ (𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))) ↔ ∃𝑥𝑦(𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥}) ∧ (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}))))
183 df-rex 2902 . . . . . . . . 9 (∃𝑦𝑉 ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) ↔ ∃𝑦(𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉))))
184183rexbii 3023 . . . . . . . 8 (∃𝑥𝑉𝑦𝑉 ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) ↔ ∃𝑥𝑉𝑦(𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉))))
185 df-rex 2902 . . . . . . . 8 (∃𝑥𝑉𝑦(𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉))) ↔ ∃𝑥(𝑥𝑉 ∧ ∃𝑦(𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))))
186 19.42v 1905 . . . . . . . . . 10 (∃𝑦(𝑥𝑉 ∧ (𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))) ↔ (𝑥𝑉 ∧ ∃𝑦(𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))))
187186bicomi 213 . . . . . . . . 9 ((𝑥𝑉 ∧ ∃𝑦(𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))) ↔ ∃𝑦(𝑥𝑉 ∧ (𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))))
188187exbii 1764 . . . . . . . 8 (∃𝑥(𝑥𝑉 ∧ ∃𝑦(𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))) ↔ ∃𝑥𝑦(𝑥𝑉 ∧ (𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))))
189184, 185, 1883bitri 285 . . . . . . 7 (∃𝑥𝑉𝑦𝑉 ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) ↔ ∃𝑥𝑦(𝑥𝑉 ∧ (𝑦𝑉 ∧ ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)))))
190 df-rex 2902 . . . . . . . 8 (∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩} ↔ ∃𝑥(𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ ∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}))
191 r19.42v 3073 . . . . . . . . . 10 (∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})(𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}) ↔ (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ ∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}))
192 df-rex 2902 . . . . . . . . . 10 (∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})(𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}) ↔ ∃𝑦(𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥}) ∧ (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩})))
193191, 192bitr3i 265 . . . . . . . . 9 ((𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ ∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}) ↔ ∃𝑦(𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥}) ∧ (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩})))
194193exbii 1764 . . . . . . . 8 (∃𝑥(𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ ∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}) ↔ ∃𝑥𝑦(𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥}) ∧ (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩})))
195190, 194bitri 263 . . . . . . 7 (∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩} ↔ ∃𝑥𝑦(𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥}) ∧ (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∧ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩})))
196182, 189, 1953bitr4g 302 . . . . . 6 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (∃𝑥𝑉𝑦𝑉 ((∃𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁) ∧ 𝑧 ∈ ((𝑉 × 𝑉) × 𝑉)) ↔ ∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}))
19734, 196syl5bb 271 . . . . 5 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((𝑧 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨𝑥, 𝑚, 𝑦⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑦} ∈ ran 𝐸)) ∧ (2nd ‘(1st𝑧)) = 𝑁)) ↔ ∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}))
19819, 29, 1973bitrd 293 . . . 4 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑧 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑁)} ↔ ∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}))
199 vex 3176 . . . . . . 7 𝑧 ∈ V
200 eleq1 2676 . . . . . . . 8 (𝑝 = 𝑧 → (𝑝 ∈ {⟨𝑥, 𝑁, 𝑦⟩} ↔ 𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}))
2012002rexbidv 3039 . . . . . . 7 (𝑝 = 𝑧 → (∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑝 ∈ {⟨𝑥, 𝑁, 𝑦⟩} ↔ ∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩}))
202199, 201elab 3319 . . . . . 6 (𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑝 ∈ {⟨𝑥, 𝑁, 𝑦⟩}} ↔ ∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩})
203202bicomi 213 . . . . 5 (∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩} ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑝 ∈ {⟨𝑥, 𝑁, 𝑦⟩}})
204203a1i 11 . . . 4 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑧 ∈ {⟨𝑥, 𝑁, 𝑦⟩} ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑝 ∈ {⟨𝑥, 𝑁, 𝑦⟩}}))
20512, 198, 2043bitrd 293 . . 3 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑧 ∈ (𝑀𝑁) ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑝 ∈ {⟨𝑥, 𝑁, 𝑦⟩}}))
206205eqrdv 2608 . 2 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑀𝑁) = {𝑝 ∣ ∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑝 ∈ {⟨𝑥, 𝑁, 𝑦⟩}})
207 dfiunv2 4492 . 2 𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) 𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥}){⟨𝑥, 𝑁, 𝑦⟩} = {𝑝 ∣ ∃𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁)∃𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥})𝑝 ∈ {⟨𝑥, 𝑁, 𝑦⟩}}
208206, 207syl6eqr 2662 1 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑀𝑁) = 𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) 𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥}){⟨𝑥, 𝑁, 𝑦⟩})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537  {csn 4125  {cpr 4127  ⟨cop 4131  ⟨cotp 4133  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ran crn 5039  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Fincfn 7841   USGrph cusg 25859   Neighbors cnbgra 25946   2SPathsOt c2spthot 26383   2SPathOnOt c2pthonot 26384 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-nbgra 25949  df-wlk 26036  df-trail 26037  df-pth 26038  df-spth 26039  df-wlkon 26042  df-spthon 26045  df-2wlkonot 26385  df-2spthonot 26387  df-2spthsot 26388 This theorem is referenced by:  usgreghash2spotv  26593
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