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Theorem frgrncvvdeqlem8 28084
Description: Lemma 8 for frgrncvvdeq 28087. This corresponds to statement 2 in [Huneke] p. 1: "The map is one-to-one since z in N(x) is uniquely determined as the common neighbor of x and a(x)". (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.) (Revised by AV, 30-Dec-2021.)
Hypotheses
Ref Expression
frgrncvvdeq.v1 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.e 𝐸 = (Edg‘𝐺)
frgrncvvdeq.nx 𝐷 = (𝐺 NeighbVtx 𝑋)
frgrncvvdeq.ny 𝑁 = (𝐺 NeighbVtx 𝑌)
frgrncvvdeq.x (𝜑𝑋𝑉)
frgrncvvdeq.y (𝜑𝑌𝑉)
frgrncvvdeq.ne (𝜑𝑋𝑌)
frgrncvvdeq.xy (𝜑𝑌𝐷)
frgrncvvdeq.f (𝜑𝐺 ∈ FriendGraph )
frgrncvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
Assertion
Ref Expression
frgrncvvdeqlem8 (𝜑𝐴:𝐷1-1𝑁)
Distinct variable groups:   𝑦,𝐸   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝑥,𝑦,𝑁   𝑥,𝐷   𝑥,𝑁   𝜑,𝑥   𝑦,𝐷   𝑥,𝐸
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑥,𝑦)   𝐺(𝑥)   𝑉(𝑥)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrncvvdeqlem8
Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrncvvdeq.v1 . . 3 𝑉 = (Vtx‘𝐺)
2 frgrncvvdeq.e . . 3 𝐸 = (Edg‘𝐺)
3 frgrncvvdeq.nx . . 3 𝐷 = (𝐺 NeighbVtx 𝑋)
4 frgrncvvdeq.ny . . 3 𝑁 = (𝐺 NeighbVtx 𝑌)
5 frgrncvvdeq.x . . 3 (𝜑𝑋𝑉)
6 frgrncvvdeq.y . . 3 (𝜑𝑌𝑉)
7 frgrncvvdeq.ne . . 3 (𝜑𝑋𝑌)
8 frgrncvvdeq.xy . . 3 (𝜑𝑌𝐷)
9 frgrncvvdeq.f . . 3 (𝜑𝐺 ∈ FriendGraph )
10 frgrncvvdeq.a . . 3 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10frgrncvvdeqlem4 28080 . 2 (𝜑𝐴:𝐷𝑁)
12 simpr 487 . . 3 ((𝜑𝐴:𝐷𝑁) → 𝐴:𝐷𝑁)
13 ffvelrn 6848 . . . . . . . . 9 ((𝐴:𝐷𝑁𝑢𝐷) → (𝐴𝑢) ∈ 𝑁)
1413ad2ant2lr 746 . . . . . . . 8 (((𝜑𝐴:𝐷𝑁) ∧ (𝑢𝐷𝑤𝐷)) → (𝐴𝑢) ∈ 𝑁)
1514adantr 483 . . . . . . 7 ((((𝜑𝐴:𝐷𝑁) ∧ (𝑢𝐷𝑤𝐷)) ∧ (𝐴𝑢) = (𝐴𝑤)) → (𝐴𝑢) ∈ 𝑁)
161, 2, 3, 4, 5, 6, 7, 8, 9, 10frgrncvvdeqlem1 28077 . . . . . . . . . . 11 (𝜑𝑋𝑁)
17 preq1 4668 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑢 → {𝑥, 𝑦} = {𝑢, 𝑦})
1817eleq1d 2897 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑢 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑢, 𝑦} ∈ 𝐸))
1918riotabidv 7115 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑢 → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = (𝑦𝑁 {𝑢, 𝑦} ∈ 𝐸))
2019cbvmptv 5168 . . . . . . . . . . . . . . . . . 18 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑢𝐷 ↦ (𝑦𝑁 {𝑢, 𝑦} ∈ 𝐸))
2110, 20eqtri 2844 . . . . . . . . . . . . . . . . 17 𝐴 = (𝑢𝐷 ↦ (𝑦𝑁 {𝑢, 𝑦} ∈ 𝐸))
221, 2, 3, 4, 5, 6, 7, 8, 9, 21frgrncvvdeqlem6 28082 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝐷) → {𝑢, (𝐴𝑢)} ∈ 𝐸)
23 preq1 4668 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → {𝑥, 𝑦} = {𝑤, 𝑦})
2423eleq1d 2897 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑤 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑤, 𝑦} ∈ 𝐸))
2524riotabidv 7115 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑤 → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = (𝑦𝑁 {𝑤, 𝑦} ∈ 𝐸))
2625cbvmptv 5168 . . . . . . . . . . . . . . . . . 18 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑤𝐷 ↦ (𝑦𝑁 {𝑤, 𝑦} ∈ 𝐸))
2710, 26eqtri 2844 . . . . . . . . . . . . . . . . 17 𝐴 = (𝑤𝐷 ↦ (𝑦𝑁 {𝑤, 𝑦} ∈ 𝐸))
281, 2, 3, 4, 5, 6, 7, 8, 9, 27frgrncvvdeqlem6 28082 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐷) → {𝑤, (𝐴𝑤)} ∈ 𝐸)
2922, 28anim12dan 620 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸))
30 preq2 4669 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑤) = (𝐴𝑢) → {𝑤, (𝐴𝑤)} = {𝑤, (𝐴𝑢)})
3130eleq1d 2897 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑤) = (𝐴𝑢) → ({𝑤, (𝐴𝑤)} ∈ 𝐸 ↔ {𝑤, (𝐴𝑢)} ∈ 𝐸))
3231anbi2d 630 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑤) = (𝐴𝑢) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸) ↔ ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸)))
3332eqcoms 2829 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑢) = (𝐴𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸) ↔ ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸)))
3433biimpa 479 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑢) = (𝐴𝑤) ∧ ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸)) → ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸))
35 df-ne 3017 . . . . . . . . . . . . . . . . . . 19 (𝑢𝑤 ↔ ¬ 𝑢 = 𝑤)
362, 3frgrnbnb 28071 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ FriendGraph ∧ (𝑢𝐷𝑤𝐷) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝐴𝑢) = 𝑋))
379, 36syl3an1 1159 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑢𝐷𝑤𝐷) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝐴𝑢) = 𝑋))
38373expa 1114 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑢𝐷𝑤𝐷)) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝐴𝑢) = 𝑋))
39 df-nel 3124 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑋𝑁 ↔ ¬ 𝑋𝑁)
40 eleq1 2900 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴𝑢) = 𝑋 → ((𝐴𝑢) ∈ 𝑁𝑋𝑁))
4140biimpa 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐴𝑢) = 𝑋 ∧ (𝐴𝑢) ∈ 𝑁) → 𝑋𝑁)
4241pm2.24d 154 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐴𝑢) = 𝑋 ∧ (𝐴𝑢) ∈ 𝑁) → (¬ 𝑋𝑁𝑢 = 𝑤))
4342expcom 416 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝑢) ∈ 𝑁 → ((𝐴𝑢) = 𝑋 → (¬ 𝑋𝑁𝑢 = 𝑤)))
4443com13 88 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑋𝑁 → ((𝐴𝑢) = 𝑋 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))
4539, 44sylbi 219 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑋𝑁 → ((𝐴𝑢) = 𝑋 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))
4645com12 32 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑢) = 𝑋 → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))
4738, 46syl6 35 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑢𝐷𝑤𝐷)) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))
4847expcom 416 . . . . . . . . . . . . . . . . . . . 20 (𝑢𝑤 → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
4948com23 86 . . . . . . . . . . . . . . . . . . 19 (𝑢𝑤 → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
5035, 49sylbir 237 . . . . . . . . . . . . . . . . . 18 𝑢 = 𝑤 → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
5134, 50syl5com 31 . . . . . . . . . . . . . . . . 17 (((𝐴𝑢) = (𝐴𝑤) ∧ ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸)) → (¬ 𝑢 = 𝑤 → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
5251expcom 416 . . . . . . . . . . . . . . . 16 (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸) → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
5352com24 95 . . . . . . . . . . . . . . 15 (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸) → ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) = (𝐴𝑤) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
5429, 53mpcom 38 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) = (𝐴𝑤) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
5554ex 415 . . . . . . . . . . . . 13 (𝜑 → ((𝑢𝐷𝑤𝐷) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) = (𝐴𝑤) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
5655com3r 87 . . . . . . . . . . . 12 𝑢 = 𝑤 → (𝜑 → ((𝑢𝐷𝑤𝐷) → ((𝐴𝑢) = (𝐴𝑤) → (𝑋𝑁 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
5756com15 101 . . . . . . . . . . 11 (𝑋𝑁 → (𝜑 → ((𝑢𝐷𝑤𝐷) → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
5816, 57mpcom 38 . . . . . . . . . 10 (𝜑 → ((𝑢𝐷𝑤𝐷) → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))))
5958expd 418 . . . . . . . . 9 (𝜑 → (𝑢𝐷 → (𝑤𝐷 → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
6059adantr 483 . . . . . . . 8 ((𝜑𝐴:𝐷𝑁) → (𝑢𝐷 → (𝑤𝐷 → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
6160imp42 429 . . . . . . 7 ((((𝜑𝐴:𝐷𝑁) ∧ (𝑢𝐷𝑤𝐷)) ∧ (𝐴𝑢) = (𝐴𝑤)) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))
6215, 61mpid 44 . . . . . 6 ((((𝜑𝐴:𝐷𝑁) ∧ (𝑢𝐷𝑤𝐷)) ∧ (𝐴𝑢) = (𝐴𝑤)) → (¬ 𝑢 = 𝑤𝑢 = 𝑤))
6362pm2.18d 127 . . . . 5 ((((𝜑𝐴:𝐷𝑁) ∧ (𝑢𝐷𝑤𝐷)) ∧ (𝐴𝑢) = (𝐴𝑤)) → 𝑢 = 𝑤)
6463ex 415 . . . 4 (((𝜑𝐴:𝐷𝑁) ∧ (𝑢𝐷𝑤𝐷)) → ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤))
6564ralrimivva 3191 . . 3 ((𝜑𝐴:𝐷𝑁) → ∀𝑢𝐷𝑤𝐷 ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤))
66 dff13 7012 . . 3 (𝐴:𝐷1-1𝑁 ↔ (𝐴:𝐷𝑁 ∧ ∀𝑢𝐷𝑤𝐷 ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤)))
6712, 65, 66sylanbrc 585 . 2 ((𝜑𝐴:𝐷𝑁) → 𝐴:𝐷1-1𝑁)
6811, 67mpdan 685 1 (𝜑𝐴:𝐷1-1𝑁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  wne 3016  wnel 3123  wral 3138  {cpr 4568  cmpt 5145  wf 6350  1-1wf1 6351  cfv 6354  crio 7112  (class class class)co 7155  Vtxcvtx 26780  Edgcedg 26831   NeighbVtx cnbgr 27113   FriendGraph cfrgr 28036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-2o 8102  df-oadd 8105  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-dju 9329  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-2 11699  df-n0 11897  df-xnn0 11967  df-z 11981  df-uz 12243  df-fz 12892  df-hash 13690  df-edg 26832  df-upgr 26866  df-umgr 26867  df-usgr 26935  df-nbgr 27114  df-frgr 28037
This theorem is referenced by:  frgrncvvdeqlem10  28086
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