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Theorem nbgrel 27133
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 27128 . . 3 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋𝑉)
32pm4.71ri 564 . 2 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ (𝑋𝑉𝑁 ∈ (𝐺 NeighbVtx 𝑋)))
4 nbgrel.e . . . . . . 7 𝐸 = (Edg‘𝐺)
51, 4nbgrval 27129 . . . . . 6 (𝑋𝑉 → (𝐺 NeighbVtx 𝑋) = {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒})
65eleq2d 2878 . . . . 5 (𝑋𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒}))
7 preq2 4633 . . . . . . . . 9 (𝑛 = 𝑁 → {𝑋, 𝑛} = {𝑋, 𝑁})
87sseq1d 3949 . . . . . . . 8 (𝑛 = 𝑁 → ({𝑋, 𝑛} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ 𝑒))
98rexbidv 3259 . . . . . . 7 (𝑛 = 𝑁 → (∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
109elrab 3631 . . . . . 6 (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ (𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
11 eldifsn 4683 . . . . . . 7 (𝑁 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑁𝑉𝑁𝑋))
1211anbi1i 626 . . . . . 6 ((𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
1310, 12bitri 278 . . . . 5 (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
146, 13syl6bb 290 . . . 4 (𝑋𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
1514pm5.32i 578 . . 3 ((𝑋𝑉𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ (𝑋𝑉 ∧ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
16 df-3an 1086 . . . 4 (((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
17 anass 472 . . . . . 6 (((𝑋𝑉𝑁𝑉) ∧ 𝑁𝑋) ↔ (𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)))
18 ancom 464 . . . . . . 7 ((𝑋𝑉𝑁𝑉) ↔ (𝑁𝑉𝑋𝑉))
1918anbi1i 626 . . . . . 6 (((𝑋𝑉𝑁𝑉) ∧ 𝑁𝑋) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋))
2017, 19bitr3i 280 . . . . 5 ((𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋))
2120anbi1i 626 . . . 4 (((𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
22 anass 472 . . . 4 (((𝑋𝑉 ∧ (𝑁𝑉𝑁𝑋)) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (𝑋𝑉 ∧ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
2316, 21, 223bitr2ri 303 . . 3 ((𝑋𝑉 ∧ ((𝑁𝑉𝑁𝑋) ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒)) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
2415, 23bitri 278 . 2 ((𝑋𝑉𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
253, 24bitri 278 1 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁𝑉𝑋𝑉) ∧ 𝑁𝑋 ∧ ∃𝑒𝐸 {𝑋, 𝑁} ⊆ 𝑒))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wrex 3110  {crab 3113  cdif 3881  wss 3884  {csn 4528  {cpr 4530  cfv 6328  (class class class)co 7139  Vtxcvtx 26792  Edgcedg 26843   NeighbVtx cnbgr 27125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-nbgr 27126
This theorem is referenced by:  nbgrisvtx  27134  nbgr2vtx1edg  27143  nbuhgr2vtx1edgblem  27144  nbuhgr2vtx1edgb  27145  nbgrsym  27156  isuvtx  27188  iscplgredg  27210  cusgrexi  27236  structtocusgr  27239
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