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Theorem frgrncvvdeqlem8 30325
Description: Lemma 8 for frgrncvvdeq 30328. 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 30321 . 2 (𝜑𝐴:𝐷𝑁)
12 simpr 484 . . 3 ((𝜑𝐴:𝐷𝑁) → 𝐴:𝐷𝑁)
13 ffvelcdm 7101 . . . . . . . . 9 ((𝐴:𝐷𝑁𝑢𝐷) → (𝐴𝑢) ∈ 𝑁)
1413ad2ant2lr 748 . . . . . . . 8 (((𝜑𝐴:𝐷𝑁) ∧ (𝑢𝐷𝑤𝐷)) → (𝐴𝑢) ∈ 𝑁)
1514adantr 480 . . . . . . 7 ((((𝜑𝐴:𝐷𝑁) ∧ (𝑢𝐷𝑤𝐷)) ∧ (𝐴𝑢) = (𝐴𝑤)) → (𝐴𝑢) ∈ 𝑁)
161, 2, 3, 4, 5, 6, 7, 8, 9, 10frgrncvvdeqlem1 30318 . . . . . . . . . . 11 (𝜑𝑋𝑁)
17 preq1 4733 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑢 → {𝑥, 𝑦} = {𝑢, 𝑦})
1817eleq1d 2826 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑢 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑢, 𝑦} ∈ 𝐸))
1918riotabidv 7390 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑢 → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = (𝑦𝑁 {𝑢, 𝑦} ∈ 𝐸))
2019cbvmptv 5255 . . . . . . . . . . . . . . . . . 18 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑢𝐷 ↦ (𝑦𝑁 {𝑢, 𝑦} ∈ 𝐸))
2110, 20eqtri 2765 . . . . . . . . . . . . . . . . 17 𝐴 = (𝑢𝐷 ↦ (𝑦𝑁 {𝑢, 𝑦} ∈ 𝐸))
221, 2, 3, 4, 5, 6, 7, 8, 9, 21frgrncvvdeqlem6 30323 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝐷) → {𝑢, (𝐴𝑢)} ∈ 𝐸)
23 preq1 4733 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → {𝑥, 𝑦} = {𝑤, 𝑦})
2423eleq1d 2826 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑤 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑤, 𝑦} ∈ 𝐸))
2524riotabidv 7390 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑤 → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = (𝑦𝑁 {𝑤, 𝑦} ∈ 𝐸))
2625cbvmptv 5255 . . . . . . . . . . . . . . . . . 18 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑤𝐷 ↦ (𝑦𝑁 {𝑤, 𝑦} ∈ 𝐸))
2710, 26eqtri 2765 . . . . . . . . . . . . . . . . 17 𝐴 = (𝑤𝐷 ↦ (𝑦𝑁 {𝑤, 𝑦} ∈ 𝐸))
281, 2, 3, 4, 5, 6, 7, 8, 9, 27frgrncvvdeqlem6 30323 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐷) → {𝑤, (𝐴𝑤)} ∈ 𝐸)
2922, 28anim12dan 619 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸))
30 preq2 4734 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑤) = (𝐴𝑢) → {𝑤, (𝐴𝑤)} = {𝑤, (𝐴𝑢)})
3130eleq1d 2826 . . . . . . . . . . . . . . . . . . . . 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 2745 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑢) = (𝐴𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸) ↔ ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸)))
3433biimpa 476 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑢) = (𝐴𝑤) ∧ ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸)) → ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸))
35 df-ne 2941 . . . . . . . . . . . . . . . . . . 19 (𝑢𝑤 ↔ ¬ 𝑢 = 𝑤)
362, 3frgrnbnb 30312 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ FriendGraph ∧ (𝑢𝐷𝑤𝐷) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝐴𝑢) = 𝑋))
379, 36syl3an1 1164 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑢𝐷𝑤𝐷) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝐴𝑢) = 𝑋))
38373expa 1119 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑢𝐷𝑤𝐷)) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝐴𝑢) = 𝑋))
39 df-nel 3047 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑋𝑁 ↔ ¬ 𝑋𝑁)
40 eleq1 2829 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐴𝑢) = 𝑋 → ((𝐴𝑢) ∈ 𝑁𝑋𝑁))
4140biimpa 476 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐴𝑢) = 𝑋 ∧ (𝐴𝑢) ∈ 𝑁) → 𝑋𝑁)
4241pm2.24d 151 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐴𝑢) = 𝑋 ∧ (𝐴𝑢) ∈ 𝑁) → (¬ 𝑋𝑁𝑢 = 𝑤))
4342expcom 413 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝑢) ∈ 𝑁 → ((𝐴𝑢) = 𝑋 → (¬ 𝑋𝑁𝑢 = 𝑤)))
4443com13 88 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑋𝑁 → ((𝐴𝑢) = 𝑋 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤)))
4539, 44sylbi 217 . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . 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 415 . . . . . . . . 9 (𝜑 → (𝑢𝐷 → (𝑤𝐷 → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
6059adantr 480 . . . . . . . 8 ((𝜑𝐴:𝐷𝑁) → (𝑢𝐷 → (𝑤𝐷 → ((𝐴𝑢) = (𝐴𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴𝑢) ∈ 𝑁𝑢 = 𝑤))))))
6160imp42 426 . . . . . . 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 412 . . . 4 (((𝜑𝐴:𝐷𝑁) ∧ (𝑢𝐷𝑤𝐷)) → ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤))
6564ralrimivva 3202 . . 3 ((𝜑𝐴:𝐷𝑁) → ∀𝑢𝐷𝑤𝐷 ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤))
66 dff13 7275 . . 3 (𝐴:𝐷1-1𝑁 ↔ (𝐴:𝐷𝑁 ∧ ∀𝑢𝐷𝑤𝐷 ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤)))
6712, 65, 66sylanbrc 583 . 2 ((𝜑𝐴:𝐷𝑁) → 𝐴:𝐷1-1𝑁)
6811, 67mpdan 687 1 (𝜑𝐴:𝐷1-1𝑁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wnel 3046  wral 3061  {cpr 4628  cmpt 5225  wf 6557  1-1wf1 6558  cfv 6561  crio 7387  (class class class)co 7431  Vtxcvtx 29013  Edgcedg 29064   NeighbVtx cnbgr 29349   FriendGraph cfrgr 30277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370  df-edg 29065  df-upgr 29099  df-umgr 29100  df-usgr 29168  df-nbgr 29350  df-frgr 30278
This theorem is referenced by:  frgrncvvdeqlem10  30327
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