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Theorem nbgrel 29152
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 29147 . . 3 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋𝑉)
32pm4.71ri 560 . 2 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ (𝑋𝑉𝑁 ∈ (𝐺 NeighbVtx 𝑋)))
4 nbgrel.e . . . . . . 7 𝐸 = (Edg‘𝐺)
51, 4nbgrval 29148 . . . . . 6 (𝑋𝑉 → (𝐺 NeighbVtx 𝑋) = {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒})
65eleq2d 2815 . . . . 5 (𝑋𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒}))
7 preq2 4739 . . . . . . . . 9 (𝑛 = 𝑁 → {𝑋, 𝑛} = {𝑋, 𝑁})
87sseq1d 4011 . . . . . . . 8 (𝑛 = 𝑁 → ({𝑋, 𝑛} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ 𝑒))
98rexbidv 3175 . . . . . . 7 (𝑛 = 𝑁 → (∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
109elrab 3682 . . . . . 6 (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ (𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
11 eldifsn 4791 . . . . . . 7 (𝑁 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑁𝑉𝑁𝑋))
1211anbi1i 623 . . . . . 6 ((𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
1310, 12bitri 275 . . . . 5 (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
146, 13bitrdi 287 . . . 4 (𝑋𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
1514pm5.32i 574 . . 3 ((𝑋𝑉𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ (𝑋𝑉 ∧ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
16 df-3an 1087 . . . 4 (((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
17 anass 468 . . . . . 6 (((𝑋𝑉𝑁𝑉) ∧ 𝑁𝑋) ↔ (𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)))
18 ancom 460 . . . . . . 7 ((𝑋𝑉𝑁𝑉) ↔ (𝑁𝑉𝑋𝑉))
1918anbi1i 623 . . . . . 6 (((𝑋𝑉𝑁𝑉) ∧ 𝑁𝑋) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋))
2017, 19bitr3i 277 . . . . 5 ((𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋))
2120anbi1i 623 . . . 4 (((𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
22 anass 468 . . . 4 (((𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (𝑋𝑉 ∧ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
2316, 21, 223bitr2ri 300 . . 3 ((𝑋𝑉 ∧ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
2415, 23bitri 275 . 2 ((𝑋𝑉𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
253, 24bitri 275 1 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wne 2937  wrex 3067  {crab 3429  cdif 3944  wss 3947  {csn 4629  {cpr 4631  cfv 6548  (class class class)co 7420  Vtxcvtx 28808  Edgcedg 28859   NeighbVtx cnbgr 29144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-nbgr 29145
This theorem is referenced by:  nbgrisvtx  29153  nbgr2vtx1edg  29162  nbuhgr2vtx1edgblem  29163  nbuhgr2vtx1edgb  29164  nbgrsym  29175  isuvtx  29207  iscplgredg  29229  cusgrexi  29255  structtocusgr  29258
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