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Theorem nbgrel 27049
Description: Characterization of a neighbor 𝑁 of a vertex 𝑋 in a graph 𝐺. (Contributed by Alexander van der Vekens and Mario Carneiro, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Revised by AV, 12-Feb-2022.)
Hypotheses
Ref Expression
nbgrel.v 𝑉 = (Vtx‘𝐺)
nbgrel.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
nbgrel (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑋   𝑒,𝑉

Proof of Theorem nbgrel
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 nbgrel.v . . . 4 𝑉 = (Vtx‘𝐺)
21nbgrcl 27044 . . 3 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋𝑉)
32pm4.71ri 561 . 2 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ (𝑋𝑉𝑁 ∈ (𝐺 NeighbVtx 𝑋)))
4 nbgrel.e . . . . . . 7 𝐸 = (Edg‘𝐺)
51, 4nbgrval 27045 . . . . . 6 (𝑋𝑉 → (𝐺 NeighbVtx 𝑋) = {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒})
65eleq2d 2895 . . . . 5 (𝑋𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒}))
7 preq2 4662 . . . . . . . . 9 (𝑛 = 𝑁 → {𝑋, 𝑛} = {𝑋, 𝑁})
87sseq1d 3995 . . . . . . . 8 (𝑛 = 𝑁 → ({𝑋, 𝑛} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ 𝑒))
98rexbidv 3294 . . . . . . 7 (𝑛 = 𝑁 → (∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
109elrab 3677 . . . . . 6 (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ (𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
11 eldifsn 4711 . . . . . . 7 (𝑁 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑁𝑉𝑁𝑋))
1211anbi1i 623 . . . . . 6 ((𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
1310, 12bitri 276 . . . . 5 (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
146, 13syl6bb 288 . . . 4 (𝑋𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
1514pm5.32i 575 . . 3 ((𝑋𝑉𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ (𝑋𝑉 ∧ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
16 df-3an 1081 . . . 4 (((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
17 anass 469 . . . . . 6 (((𝑋𝑉𝑁𝑉) ∧ 𝑁𝑋) ↔ (𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)))
18 ancom 461 . . . . . . 7 ((𝑋𝑉𝑁𝑉) ↔ (𝑁𝑉𝑋𝑉))
1918anbi1i 623 . . . . . 6 (((𝑋𝑉𝑁𝑉) ∧ 𝑁𝑋) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋))
2017, 19bitr3i 278 . . . . 5 ((𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋))
2120anbi1i 623 . . . 4 (((𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
22 anass 469 . . . 4 (((𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (𝑋𝑉 ∧ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
2316, 21, 223bitr2ri 301 . . 3 ((𝑋𝑉 ∧ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
2415, 23bitri 276 . 2 ((𝑋𝑉𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
253, 24bitri 276 1 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wrex 3136  {crab 3139  cdif 3930  wss 3933  {csn 4557  {cpr 4559  cfv 6348  (class class class)co 7145  Vtxcvtx 26708  Edgcedg 26759   NeighbVtx cnbgr 27041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-nbgr 27042
This theorem is referenced by:  nbgrisvtx  27050  nbgr2vtx1edg  27059  nbuhgr2vtx1edgblem  27060  nbuhgr2vtx1edgb  27061  nbgrsym  27072  isuvtx  27104  iscplgredg  27126  cusgrexi  27152  structtocusgr  27155
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