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Theorem nbgrel 26278
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 26272 . . 3 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋𝑉)
32pm4.71ri 666 . 2 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ (𝑋𝑉𝑁 ∈ (𝐺 NeighbVtx 𝑋)))
4 nbgrel.e . . . . . . 7 𝐸 = (Edg‘𝐺)
51, 4nbgrval 26274 . . . . . 6 (𝑋𝑉 → (𝐺 NeighbVtx 𝑋) = {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒})
65eleq2d 2716 . . . . 5 (𝑋𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒}))
7 preq2 4301 . . . . . . . . 9 (𝑛 = 𝑁 → {𝑋, 𝑛} = {𝑋, 𝑁})
87sseq1d 3665 . . . . . . . 8 (𝑛 = 𝑁 → ({𝑋, 𝑛} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ 𝑒))
98rexbidv 3081 . . . . . . 7 (𝑛 = 𝑁 → (∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
109elrab 3396 . . . . . 6 (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ (𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
11 eldifsn 4350 . . . . . . 7 (𝑁 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑁𝑉𝑁𝑋))
1211anbi1i 731 . . . . . 6 ((𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
1310, 12bitri 264 . . . . 5 (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
146, 13syl6bb 276 . . . 4 (𝑋𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
1514pm5.32i 670 . . 3 ((𝑋𝑉𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ (𝑋𝑉 ∧ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
16 df-3an 1056 . . . 4 (((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
17 anass 682 . . . . . 6 (((𝑋𝑉𝑁𝑉) ∧ 𝑁𝑋) ↔ (𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)))
18 ancom 465 . . . . . . 7 ((𝑋𝑉𝑁𝑉) ↔ (𝑁𝑉𝑋𝑉))
1918anbi1i 731 . . . . . 6 (((𝑋𝑉𝑁𝑉) ∧ 𝑁𝑋) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋))
2017, 19bitr3i 266 . . . . 5 ((𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋))
2120anbi1i 731 . . . 4 (((𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
22 anass 682 . . . 4 (((𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (𝑋𝑉 ∧ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
2316, 21, 223bitr2ri 289 . . 3 ((𝑋𝑉 ∧ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
2415, 23bitri 264 . 2 ((𝑋𝑉𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
253, 24bitri 264 1 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wrex 2942  {crab 2945  cdif 3604  wss 3607  {csn 4210  {cpr 4212  cfv 5926  (class class class)co 6690  Vtxcvtx 25919  Edgcedg 25984   NeighbVtx cnbgr 26269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-nbgr 26270
This theorem is referenced by:  nbgrisvtx  26280  nbgr2vtx1edg  26291  nbuhgr2vtx1edgblem  26292  nbuhgr2vtx1edgb  26293  nbgrsym  26308  isuvtx  26343  isuvtxaOLD  26344  iscplgredg  26369  cusgrexi  26395  structtocusgr  26398
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