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Theorem nbgrel 26825
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 26820 . . 3 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋𝑉)
32pm4.71ri 553 . 2 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ (𝑋𝑉𝑁 ∈ (𝐺 NeighbVtx 𝑋)))
4 nbgrel.e . . . . . . 7 𝐸 = (Edg‘𝐺)
51, 4nbgrval 26821 . . . . . 6 (𝑋𝑉 → (𝐺 NeighbVtx 𝑋) = {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒})
65eleq2d 2851 . . . . 5 (𝑋𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒}))
7 preq2 4544 . . . . . . . . 9 (𝑛 = 𝑁 → {𝑋, 𝑛} = {𝑋, 𝑁})
87sseq1d 3888 . . . . . . . 8 (𝑛 = 𝑁 → ({𝑋, 𝑛} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ 𝑒))
98rexbidv 3242 . . . . . . 7 (𝑛 = 𝑁 → (∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
109elrab 3595 . . . . . 6 (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ (𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
11 eldifsn 4593 . . . . . . 7 (𝑁 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑁𝑉𝑁𝑋))
1211anbi1i 614 . . . . . 6 ((𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
1310, 12bitri 267 . . . . 5 (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
146, 13syl6bb 279 . . . 4 (𝑋𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
1514pm5.32i 567 . . 3 ((𝑋𝑉𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ (𝑋𝑉 ∧ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
16 df-3an 1070 . . . 4 (((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
17 anass 461 . . . . . 6 (((𝑋𝑉𝑁𝑉) ∧ 𝑁𝑋) ↔ (𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)))
18 ancom 453 . . . . . . 7 ((𝑋𝑉𝑁𝑉) ↔ (𝑁𝑉𝑋𝑉))
1918anbi1i 614 . . . . . 6 (((𝑋𝑉𝑁𝑉) ∧ 𝑁𝑋) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋))
2017, 19bitr3i 269 . . . . 5 ((𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋))
2120anbi1i 614 . . . 4 (((𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
22 anass 461 . . . 4 (((𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (𝑋𝑉 ∧ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
2316, 21, 223bitr2ri 292 . . 3 ((𝑋𝑉 ∧ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
2415, 23bitri 267 . 2 ((𝑋𝑉𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
253, 24bitri 267 1 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wne 2967  wrex 3089  {crab 3092  cdif 3826  wss 3829  {csn 4441  {cpr 4443  cfv 6188  (class class class)co 6976  Vtxcvtx 26484  Edgcedg 26535   NeighbVtx cnbgr 26817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-nbgr 26818
This theorem is referenced by:  nbgrisvtx  26826  nbgr2vtx1edg  26835  nbuhgr2vtx1edgblem  26836  nbuhgr2vtx1edgb  26837  nbgrsym  26848  isuvtx  26880  iscplgredg  26902  cusgrexi  26928  structtocusgr  26931
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