MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgrncvvdeqlem8 Structured version   Visualization version   GIF version

Theorem frgrncvvdeqlem8 29827
Description: Lemma 8 for frgrncvvdeq 29830. 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 29823 . 2 (𝜑𝐴:𝐷𝑁)
12 simpr 484 . . 3 ((𝜑𝐴:𝐷𝑁) → 𝐴:𝐷𝑁)
13 ffvelcdm 7083 . . . . . . . . 9 ((𝐴:𝐷𝑁𝑢𝐷) → (𝐴𝑢) ∈ 𝑁)
1413ad2ant2lr 745 . . . . . . . 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 29820 . . . . . . . . . . 11 (𝜑𝑋𝑁)
17 preq1 4737 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑢 → {𝑥, 𝑦} = {𝑢, 𝑦})
1817eleq1d 2817 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑢 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑢, 𝑦} ∈ 𝐸))
1918riotabidv 7370 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑢 → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = (𝑦𝑁 {𝑢, 𝑦} ∈ 𝐸))
2019cbvmptv 5261 . . . . . . . . . . . . . . . . . 18 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑢𝐷 ↦ (𝑦𝑁 {𝑢, 𝑦} ∈ 𝐸))
2110, 20eqtri 2759 . . . . . . . . . . . . . . . . 17 𝐴 = (𝑢𝐷 ↦ (𝑦𝑁 {𝑢, 𝑦} ∈ 𝐸))
221, 2, 3, 4, 5, 6, 7, 8, 9, 21frgrncvvdeqlem6 29825 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝐷) → {𝑢, (𝐴𝑢)} ∈ 𝐸)
23 preq1 4737 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑤 → {𝑥, 𝑦} = {𝑤, 𝑦})
2423eleq1d 2817 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑤 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑤, 𝑦} ∈ 𝐸))
2524riotabidv 7370 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑤 → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = (𝑦𝑁 {𝑤, 𝑦} ∈ 𝐸))
2625cbvmptv 5261 . . . . . . . . . . . . . . . . . 18 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑤𝐷 ↦ (𝑦𝑁 {𝑤, 𝑦} ∈ 𝐸))
2710, 26eqtri 2759 . . . . . . . . . . . . . . . . 17 𝐴 = (𝑤𝐷 ↦ (𝑦𝑁 {𝑤, 𝑦} ∈ 𝐸))
281, 2, 3, 4, 5, 6, 7, 8, 9, 27frgrncvvdeqlem6 29825 . . . . . . . . . . . . . . . 16 ((𝜑𝑤𝐷) → {𝑤, (𝐴𝑤)} ∈ 𝐸)
2922, 28anim12dan 618 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢𝐷𝑤𝐷)) → ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸))
30 preq2 4738 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑤) = (𝐴𝑢) → {𝑤, (𝐴𝑤)} = {𝑤, (𝐴𝑢)})
3130eleq1d 2817 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑤) = (𝐴𝑢) → ({𝑤, (𝐴𝑤)} ∈ 𝐸 ↔ {𝑤, (𝐴𝑢)} ∈ 𝐸))
3231anbi2d 628 . . . . . . . . . . . . . . . . . . . 20 ((𝐴𝑤) = (𝐴𝑢) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑤)} ∈ 𝐸) ↔ ({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸)))
3332eqcoms 2739 . . . . . . . . . . . . . . . . . . 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 2940 . . . . . . . . . . . . . . . . . . 19 (𝑢𝑤 ↔ ¬ 𝑢 = 𝑤)
362, 3frgrnbnb 29814 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ FriendGraph ∧ (𝑢𝐷𝑤𝐷) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝐴𝑢) = 𝑋))
379, 36syl3an1 1162 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑢𝐷𝑤𝐷) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝐴𝑢) = 𝑋))
38373expa 1117 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑢𝐷𝑤𝐷)) ∧ 𝑢𝑤) → (({𝑢, (𝐴𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴𝑢)} ∈ 𝐸) → (𝐴𝑢) = 𝑋))
39 df-nel 3046 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑋𝑁 ↔ ¬ 𝑋𝑁)
40 eleq1 2820 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . . . . . . 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 3199 . . 3 ((𝜑𝐴:𝐷𝑁) → ∀𝑢𝐷𝑤𝐷 ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤))
66 dff13 7257 . . 3 (𝐴:𝐷1-1𝑁 ↔ (𝐴:𝐷𝑁 ∧ ∀𝑢𝐷𝑤𝐷 ((𝐴𝑢) = (𝐴𝑤) → 𝑢 = 𝑤)))
6712, 65, 66sylanbrc 582 . 2 ((𝜑𝐴:𝐷𝑁) → 𝐴:𝐷1-1𝑁)
6811, 67mpdan 684 1 (𝜑𝐴:𝐷1-1𝑁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wne 2939  wnel 3045  wral 3060  {cpr 4630  cmpt 5231  wf 6539  1-1wf1 6540  cfv 6543  crio 7367  (class class class)co 7412  Vtxcvtx 28524  Edgcedg 28575   NeighbVtx cnbgr 28857   FriendGraph cfrgr 29779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-2o 8471  df-oadd 8474  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9900  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-n0 12478  df-xnn0 12550  df-z 12564  df-uz 12828  df-fz 13490  df-hash 14296  df-edg 28576  df-upgr 28610  df-umgr 28611  df-usgr 28679  df-nbgr 28858  df-frgr 29780
This theorem is referenced by:  frgrncvvdeqlem10  29829
  Copyright terms: Public domain W3C validator