Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fusgr2wsp2nb Structured version   Visualization version   GIF version

Theorem fusgr2wsp2nb 41498
 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.)
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   𝑥,𝐺,𝑦   𝑤,𝐺   𝑁,𝑎,𝑤   𝑥,𝑁,𝑦   𝑥,𝑉,𝑦   𝑤,𝑉
Allowed substitution hints:   𝑀(𝑥,𝑦,𝑤,𝑎)

Proof of Theorem fusgr2wsp2nb
Dummy variables 𝑚 𝑧 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . . . 5 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → 𝑁𝑉)
2 ovex 6577 . . . . . 6 (2 WSPathsN 𝐺) ∈ V
32rabex 4740 . . . . 5 {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁} ∈ V
4 eqeq2 2621 . . . . . . . 8 (𝑎 = 𝑁 → ((𝑤‘1) = 𝑎 ↔ (𝑤‘1) = 𝑁))
54rabbidv 3164 . . . . . . 7 (𝑎 = 𝑁 → {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎} = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁})
6 fusgreg2wsp.m . . . . . . 7 𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
75, 6fvmptg 6189 . . . . . 6 ((𝑁𝑉 ∧ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁} ∈ V) → (𝑀𝑁) = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁})
87eleq2d 2673 . . . . 5 ((𝑁𝑉 ∧ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁} ∈ V) → (𝑧 ∈ (𝑀𝑁) ↔ 𝑧 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁}))
91, 3, 8sylancl 693 . . . 4 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑧 ∈ (𝑀𝑁) ↔ 𝑧 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁}))
10 fveq1 6102 . . . . . . . 8 (𝑤 = 𝑧 → (𝑤‘1) = (𝑧‘1))
1110eqeq1d 2612 . . . . . . 7 (𝑤 = 𝑧 → ((𝑤‘1) = 𝑁 ↔ (𝑧‘1) = 𝑁))
1211elrab 3331 . . . . . 6 (𝑧 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁} ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁))
1312a1i 11 . . . . 5 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑧 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁} ↔ (𝑧 ∈ (2 WSPathsN 𝐺) ∧ (𝑧‘1) = 𝑁)))
14 2nn0 11186 . . . . . . . . 9 2 ∈ ℕ0
15 frgrhash2wsp.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
1615wspthsnwspthsnon 41122 . . . . . . . . 9 ((2 ∈ ℕ0𝐺 ∈ FinUSGraph ) → (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥𝑉𝑦𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦)))
1714, 16mpan 702 . . . . . . . 8 (𝐺 ∈ FinUSGraph → (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥𝑉𝑦𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦)))
1817adantr 480 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥𝑉𝑦𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦)))
19 fusgrusgr 40541 . . . . . . . . . 10 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
2019adantr 480 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → 𝐺 ∈ USGraph )
21 eqid 2610 . . . . . . . . . 10 (Edg‘𝐺) = (Edg‘𝐺)
2215, 21usgr2wspthon 41168 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ (𝑥𝑉𝑦𝑉)) → (𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
2320, 22sylan 487 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (𝑥𝑉𝑦𝑉)) → (𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
24232rexbidva 3038 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (∃𝑥𝑉𝑦𝑉 𝑧 ∈ (𝑥(2 WSPathsNOn 𝐺)𝑦) ↔ ∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
2518, 24bitrd 267 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑧 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
2625anbi1d 737 . . . . 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)))
27 19.41vv 1902 . . . . . . 7 (∃𝑥𝑦((𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) ∧ (𝑧‘1) = 𝑁) ↔ (∃𝑥𝑦(𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) ∧ (𝑧‘1) = 𝑁))
28 velsn 4141 . . . . . . . . . . . . 13 (𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)
2928bicomi 213 . . . . . . . . . . . 12 (𝑧 = ⟨“𝑥𝑁𝑦”⟩ ↔ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})
3029anbi2i 726 . . . . . . . . . . 11 (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) ↔ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
3130anbi2i 726 . . . . . . . . . 10 ((({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) ↔ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})))
3231a1i 11 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) ↔ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))))
33 fveq1 6102 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = (⟨“𝑥𝑚𝑦”⟩‘1))
34 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑚 ∈ V
35 s3fv1 13487 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑚 ∈ V → (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚)
3634, 35ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (⟨“𝑥𝑚𝑦”⟩‘1) = 𝑚
3733, 36syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑧‘1) = 𝑚)
3837eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → ((𝑧‘1) = 𝑁𝑚 = 𝑁))
3938biimpd 218 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → ((𝑧‘1) = 𝑁𝑚 = 𝑁))
4039adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) → ((𝑧‘1) = 𝑁𝑚 = 𝑁))
4140com12 32 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧‘1) = 𝑁 → ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) → 𝑚 = 𝑁))
4241ad2antlr 759 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁) ∧ 𝑚𝑉) → ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) → 𝑚 = 𝑁))
4342imp 444 . . . . . . . . . . . . . . . . . . . 20 (((((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁) ∧ 𝑚𝑉) ∧ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦)) → 𝑚 = 𝑁)
44 preq1 4212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑚 = 𝑁 → {𝑚, 𝑦} = {𝑁, 𝑦})
45 prcom 4211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 {𝑁, 𝑦} = {𝑦, 𝑁}
4644, 45syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑚 = 𝑁 → {𝑚, 𝑦} = {𝑦, 𝑁})
4746eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑚 = 𝑁 → ({𝑚, 𝑦} ∈ (Edg‘𝐺) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
4847biimpd 218 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑚 = 𝑁 → ({𝑚, 𝑦} ∈ (Edg‘𝐺) → {𝑦, 𝑁} ∈ (Edg‘𝐺)))
4948adantld 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑚 = 𝑁 → (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)) → {𝑦, 𝑁} ∈ (Edg‘𝐺)))
5049imp 444 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑚 = 𝑁 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → {𝑦, 𝑁} ∈ (Edg‘𝐺))
51503adant2 1073 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 = 𝑁 ∧ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → {𝑦, 𝑁} ∈ (Edg‘𝐺))
52 nesym 2838 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥𝑦 ↔ ¬ 𝑦 = 𝑥)
5352biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥𝑦 → ¬ 𝑦 = 𝑥)
5453adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) → ¬ 𝑦 = 𝑥)
55543ad2ant2 1076 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 = 𝑁 ∧ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → ¬ 𝑦 = 𝑥)
5651, 55jca 553 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 = 𝑁 ∧ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥))
57 preq2 4213 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑚 = 𝑁 → {𝑥, 𝑚} = {𝑥, 𝑁})
5857eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑚 = 𝑁 → ({𝑥, 𝑚} ∈ (Edg‘𝐺) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
5958biimpcd 238 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({𝑥, 𝑚} ∈ (Edg‘𝐺) → (𝑚 = 𝑁 → {𝑥, 𝑁} ∈ (Edg‘𝐺)))
6059adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)) → (𝑚 = 𝑁 → {𝑥, 𝑁} ∈ (Edg‘𝐺)))
6160impcom 445 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 = 𝑁 ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → {𝑥, 𝑁} ∈ (Edg‘𝐺))
62613adant2 1073 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 = 𝑁 ∧ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → {𝑥, 𝑁} ∈ (Edg‘𝐺))
63 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑚 = 𝑁𝑥 = 𝑥)
64 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑚 = 𝑁𝑚 = 𝑁)
65 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑚 = 𝑁𝑦 = 𝑦)
6663, 64, 65s3eqd 13460 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑚 = 𝑁 → ⟨“𝑥𝑚𝑦”⟩ = ⟨“𝑥𝑁𝑦”⟩)
6766eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑚 = 𝑁 → (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ↔ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))
6867biimpcd 238 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = ⟨“𝑥𝑚𝑦”⟩ → (𝑚 = 𝑁𝑧 = ⟨“𝑥𝑁𝑦”⟩))
6968adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) → (𝑚 = 𝑁𝑧 = ⟨“𝑥𝑁𝑦”⟩))
7069impcom 445 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 = 𝑁 ∧ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦)) → 𝑧 = ⟨“𝑥𝑁𝑦”⟩)
71703adant3 1074 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 = 𝑁 ∧ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → 𝑧 = ⟨“𝑥𝑁𝑦”⟩)
7256, 62, 71jca32 556 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 = 𝑁 ∧ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))
73723exp 1256 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑁 → ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) → (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)) → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))))
7473adantld 482 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑁 → (((((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁) ∧ 𝑚𝑉) ∧ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦)) → (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)) → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))))
7543, 74mpcom 37 . . . . . . . . . . . . . . . . . . 19 (((((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁) ∧ 𝑚𝑉) ∧ (𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦)) → (({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)) → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))))
7675impr 647 . . . . . . . . . . . . . . . . . 18 (((((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁) ∧ 𝑚𝑉) ∧ ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))
7776a1d 25 . . . . . . . . . . . . . . . . 17 (((((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁) ∧ 𝑚𝑉) ∧ ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))))
7877ex 449 . . . . . . . . . . . . . . . 16 ((((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁) ∧ 𝑚𝑉) → (((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))))
7978rexlimdva 3013 . . . . . . . . . . . . . . 15 (((𝑥𝑉𝑦𝑉) ∧ (𝑧‘1) = 𝑁) → (∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))))
8079ex 449 . . . . . . . . . . . . . 14 ((𝑥𝑉𝑦𝑉) → ((𝑧‘1) = 𝑁 → (∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))))))
8180com23 84 . . . . . . . . . . . . 13 ((𝑥𝑉𝑦𝑉) → (∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) → ((𝑧‘1) = 𝑁 → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))))))
8281impr 647 . . . . . . . . . . . 12 ((𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) → ((𝑧‘1) = 𝑁 → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)))))
8382imp 444 . . . . . . . . . . 11 (((𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) ∧ (𝑧‘1) = 𝑁) → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))))
8483com12 32 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (((𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) ∧ (𝑧‘1) = 𝑁) → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))))
85 usgrumgr 40409 . . . . . . . . . . . . . . . . . . . . . 22 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph )
8619, 85syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝐺 ∈ FinUSGraph → 𝐺 ∈ UMGraph )
8786adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → 𝐺 ∈ UMGraph )
8887anim1i 590 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → (𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
8915, 21umgrpredgav 25813 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ UMGraph ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → (𝑥𝑉𝑁𝑉))
90 simpl 472 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝑉𝑁𝑉) → 𝑥𝑉)
9188, 89, 903syl 18 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ {𝑥, 𝑁} ∈ (Edg‘𝐺)) → 𝑥𝑉)
9291expcom 450 . . . . . . . . . . . . . . . . 17 ({𝑥, 𝑁} ∈ (Edg‘𝐺) → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → 𝑥𝑉))
9392adantr 480 . . . . . . . . . . . . . . . 16 (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → 𝑥𝑉))
9493com12 32 . . . . . . . . . . . . . . 15 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → 𝑥𝑉))
9594adantld 482 . . . . . . . . . . . . . 14 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → 𝑥𝑉))
9695imp 444 . . . . . . . . . . . . 13 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))) → 𝑥𝑉)
9787anim1i 590 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → (𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
9815, 21umgrpredgav 25813 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ UMGraph ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → (𝑦𝑉𝑁𝑉))
99 simpl 472 . . . . . . . . . . . . . . . . . 18 ((𝑦𝑉𝑁𝑉) → 𝑦𝑉)
10097, 98, 993syl 18 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ {𝑦, 𝑁} ∈ (Edg‘𝐺)) → 𝑦𝑉)
101100expcom 450 . . . . . . . . . . . . . . . 16 ({𝑦, 𝑁} ∈ (Edg‘𝐺) → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → 𝑦𝑉))
102101adantr 480 . . . . . . . . . . . . . . 15 (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → 𝑦𝑉))
103102adantr 480 . . . . . . . . . . . . . 14 ((({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → 𝑦𝑉))
104103impcom 445 . . . . . . . . . . . . 13 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))) → 𝑦𝑉)
10567biimprd 237 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑁 → (𝑧 = ⟨“𝑥𝑁𝑦”⟩ → 𝑧 = ⟨“𝑥𝑚𝑦”⟩))
106105adantld 482 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑁 → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → 𝑧 = ⟨“𝑥𝑚𝑦”⟩))
107106adantld 482 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑁 → ((({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → 𝑧 = ⟨“𝑥𝑚𝑦”⟩))
108107imp 444 . . . . . . . . . . . . . . . . . 18 ((𝑚 = 𝑁 ∧ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))) → 𝑧 = ⟨“𝑥𝑚𝑦”⟩)
10952bicomi 213 . . . . . . . . . . . . . . . . . . . . 21 𝑦 = 𝑥𝑥𝑦)
110109biimpi 205 . . . . . . . . . . . . . . . . . . . 20 𝑦 = 𝑥𝑥𝑦)
111110ad2antlr 759 . . . . . . . . . . . . . . . . . . 19 ((({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → 𝑥𝑦)
112111adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑚 = 𝑁 ∧ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))) → 𝑥𝑦)
11347biimprd 237 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑁 → ({𝑦, 𝑁} ∈ (Edg‘𝐺) → {𝑚, 𝑦} ∈ (Edg‘𝐺)))
114113adantrd 483 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑁 → (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) → {𝑚, 𝑦} ∈ (Edg‘𝐺)))
11558biimprd 237 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑁 → ({𝑥, 𝑁} ∈ (Edg‘𝐺) → {𝑥, 𝑚} ∈ (Edg‘𝐺)))
116115adantrd 483 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑁 → (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → {𝑥, 𝑚} ∈ (Edg‘𝐺)))
117114, 116anim12d 584 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑁 → ((({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → ({𝑚, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑥, 𝑚} ∈ (Edg‘𝐺))))
118117imp 444 . . . . . . . . . . . . . . . . . . 19 ((𝑚 = 𝑁 ∧ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))) → ({𝑚, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑥, 𝑚} ∈ (Edg‘𝐺)))
119118ancomd 466 . . . . . . . . . . . . . . . . . 18 ((𝑚 = 𝑁 ∧ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))) → ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))
120108, 112, 119jca31 555 . . . . . . . . . . . . . . . . 17 ((𝑚 = 𝑁 ∧ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))) → ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))
121120ex 449 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑁 → ((({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
122121adantl 481 . . . . . . . . . . . . . . 15 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ 𝑚 = 𝑁) → ((({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
1231, 122rspcimedv 3284 . . . . . . . . . . . . . 14 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
124123imp 444 . . . . . . . . . . . . 13 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))) → ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))
12596, 104, 124jca32 556 . . . . . . . . . . . 12 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))) → (𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))))
126 fveq1 6102 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ⟨“𝑥𝑁𝑦”⟩ → (𝑧‘1) = (⟨“𝑥𝑁𝑦”⟩‘1))
127 s3fv1 13487 . . . . . . . . . . . . . . . . . . 19 (𝑁𝑉 → (⟨“𝑥𝑁𝑦”⟩‘1) = 𝑁)
128126, 127sylan9eqr 2666 . . . . . . . . . . . . . . . . . 18 ((𝑁𝑉𝑧 = ⟨“𝑥𝑁𝑦”⟩) → (𝑧‘1) = 𝑁)
129128expcom 450 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨“𝑥𝑁𝑦”⟩ → (𝑁𝑉 → (𝑧‘1) = 𝑁))
130129adantl 481 . . . . . . . . . . . . . . . 16 (({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩) → (𝑁𝑉 → (𝑧‘1) = 𝑁))
131130adantl 481 . . . . . . . . . . . . . . 15 ((({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → (𝑁𝑉 → (𝑧‘1) = 𝑁))
132131com12 32 . . . . . . . . . . . . . 14 (𝑁𝑉 → ((({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → (𝑧‘1) = 𝑁))
133132adantl 481 . . . . . . . . . . . . 13 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → (𝑧‘1) = 𝑁))
134133imp 444 . . . . . . . . . . . 12 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))) → (𝑧‘1) = 𝑁)
135125, 134jca 553 . . . . . . . . . . 11 (((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) ∧ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))) → ((𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) ∧ (𝑧‘1) = 𝑁))
136135ex 449 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩)) → ((𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) ∧ (𝑧‘1) = 𝑁)))
13784, 136impbid 201 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (((𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) ∧ (𝑧‘1) = 𝑁) ↔ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 = ⟨“𝑥𝑁𝑦”⟩))))
138 eldif 3550 . . . . . . . . . . 11 (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}))
13921nbusgreledg 40575 . . . . . . . . . . . . . 14 (𝐺 ∈ USGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
14019, 139syl 17 . . . . . . . . . . . . 13 (𝐺 ∈ FinUSGraph → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
141140adantr 480 . . . . . . . . . . . 12 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑦 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑦, 𝑁} ∈ (Edg‘𝐺)))
142 velsn 4141 . . . . . . . . . . . . . 14 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
143142a1i 11 . . . . . . . . . . . . 13 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥))
144143notbid 307 . . . . . . . . . . . 12 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (¬ 𝑦 ∈ {𝑥} ↔ ¬ 𝑦 = 𝑥))
145141, 144anbi12d 743 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((𝑦 ∈ (𝐺 NeighbVtx 𝑁) ∧ ¬ 𝑦 ∈ {𝑥}) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
146138, 145syl5bb 271 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ↔ ({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥)))
14721nbusgreledg 40575 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
14819, 147syl 17 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
149148adantr 480 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ↔ {𝑥, 𝑁} ∈ (Edg‘𝐺)))
150149anbi1d 737 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}) ↔ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})))
151146, 150anbi12d 743 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ∧ (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})) ↔ (({𝑦, 𝑁} ∈ (Edg‘𝐺) ∧ ¬ 𝑦 = 𝑥) ∧ ({𝑥, 𝑁} ∈ (Edg‘𝐺) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))))
15232, 137, 1513bitr4d 299 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (((𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) ∧ (𝑧‘1) = 𝑁) ↔ (𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ∧ (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))))
1531522exbidv 1839 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (∃𝑥𝑦((𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) ∧ (𝑧‘1) = 𝑁) ↔ ∃𝑥𝑦(𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ∧ (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))))
15427, 153syl5bbr 273 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → ((∃𝑥𝑦(𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) ∧ (𝑧‘1) = 𝑁) ↔ ∃𝑥𝑦(𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ∧ (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))))
155 df-rex 2902 . . . . . . . . 9 (∃𝑦𝑉𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ∃𝑦(𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
156155rexbii 3023 . . . . . . . 8 (∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ∃𝑥𝑉𝑦(𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))))
157 df-rex 2902 . . . . . . . 8 (∃𝑥𝑉𝑦(𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺)))) ↔ ∃𝑥(𝑥𝑉 ∧ ∃𝑦(𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))))
158 19.42v 1905 . . . . . . . . . 10 (∃𝑦(𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) ↔ (𝑥𝑉 ∧ ∃𝑦(𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))))
159158bicomi 213 . . . . . . . . 9 ((𝑥𝑉 ∧ ∃𝑦(𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) ↔ ∃𝑦(𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))))
160159exbii 1764 . . . . . . . 8 (∃𝑥(𝑥𝑉 ∧ ∃𝑦(𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) ↔ ∃𝑥𝑦(𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))))
161156, 157, 1603bitri 285 . . . . . . 7 (∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ↔ ∃𝑥𝑦(𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))))
162161anbi1i 727 . . . . . 6 ((∃𝑥𝑉𝑦𝑉𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))) ∧ (𝑧‘1) = 𝑁) ↔ (∃𝑥𝑦(𝑥𝑉 ∧ (𝑦𝑉 ∧ ∃𝑚𝑉 ((𝑧 = ⟨“𝑥𝑚𝑦”⟩ ∧ 𝑥𝑦) ∧ ({𝑥, 𝑚} ∈ (Edg‘𝐺) ∧ {𝑚, 𝑦} ∈ (Edg‘𝐺))))) ∧ (𝑧‘1) = 𝑁))
163 df-rex 2902 . . . . . . 7 (∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ ∃𝑥(𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ ∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
164 r19.42v 3073 . . . . . . . . 9 (∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})(𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}) ↔ (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ ∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
165 df-rex 2902 . . . . . . . . 9 (∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})(𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}) ↔ ∃𝑦(𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ∧ (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})))
166164, 165bitr3i 265 . . . . . . . 8 ((𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ ∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}) ↔ ∃𝑦(𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ∧ (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})))
167166exbii 1764 . . . . . . 7 (∃𝑥(𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ ∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}) ↔ ∃𝑥𝑦(𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ∧ (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})))
168163, 167bitri 263 . . . . . 6 (∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ ∃𝑥𝑦(𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}) ∧ (𝑥 ∈ (𝐺 NeighbVtx 𝑁) ∧ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})))
169154, 162, 1683bitr4g 302 . . . . 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”⟩}))
17013, 26, 1693bitrd 293 . . . 4 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑧 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁} ↔ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
171 vex 3176 . . . . . . 7 𝑧 ∈ V
172 eleq1 2676 . . . . . . . 8 (𝑝 = 𝑧 → (𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
1731722rexbidv 3039 . . . . . . 7 (𝑝 = 𝑧 → (∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩}))
174171, 173elab 3319 . . . . . 6 (𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}} ↔ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩})
175174bicomi 213 . . . . 5 (∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑧 ∈ {⟨“𝑥𝑁𝑦”⟩} ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}})
176175a1i 11 . . . 4 ((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”⟩}}))
1779, 170, 1763bitrd 293 . . 3 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑧 ∈ (𝑀𝑁) ↔ 𝑧 ∈ {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}))
178177eqrdv 2608 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑀𝑁) = {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}})
179 dfiunv2 4492 . 2 𝑥 ∈ (𝐺 NeighbVtx 𝑁) 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){⟨“𝑥𝑁𝑦”⟩} = {𝑝 ∣ ∃𝑥 ∈ (𝐺 NeighbVtx 𝑁)∃𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥})𝑝 ∈ {⟨“𝑥𝑁𝑦”⟩}}
180178, 179syl6eqr 2662 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 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  ∪ ciun 4455   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  1c1 9816  2c2 10947  ℕ0cn0 11169  ⟨“cs3 13438  Vtxcvtx 25673   UMGraph cumgr 25748  Edgcedga 25792   USGraph cusgr 40379   FinUSGraph cfusgr 40535   NeighbVtx cnbgr 40550   WSPathsN cwwspthsn 41031   WSPathsNOn cwwspthsnon 41032 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-ac2 9168  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-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  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-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-se 4998  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-isom 5813  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-2o 7448  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-ac 8822  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-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-uhgr 25724  df-upgr 25749  df-umgr 25750  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-fusgr 40536  df-nbgr 40554  df-1wlks 40800  df-wlks 40801  df-wlkson 40802  df-trls 40901  df-trlson 40902  df-pths 40923  df-spths 40924  df-pthson 40925  df-spthson 40926  df-wwlks 41033  df-wwlksn 41034  df-wwlksnon 41035  df-wspthsn 41036  df-wspthsnon 41037 This theorem is referenced by:  fusgreghash2wspv  41499
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