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Theorem nbgrsym 27142
Description: In a graph, the neighborhood relation is symmetric: a vertex 𝑁 in a graph 𝐺 is a neighbor of a second vertex 𝐾 iff the second vertex 𝐾 is a neighbor of the first vertex 𝑁. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) (Revised by AV, 12-Feb-2022.)
Assertion
Ref Expression
nbgrsym (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 NeighbVtx 𝑁))

Proof of Theorem nbgrsym
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 ancom 464 . . 3 ((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ↔ (𝐾 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ (Vtx‘𝐺)))
2 necom 3066 . . 3 (𝑁𝐾𝐾𝑁)
3 prcom 4649 . . . . 5 {𝐾, 𝑁} = {𝑁, 𝐾}
43sseq1i 3979 . . . 4 ({𝐾, 𝑁} ⊆ 𝑒 ↔ {𝑁, 𝐾} ⊆ 𝑒)
54rexbii 3241 . . 3 (∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒)
61, 2, 53anbi123i 1152 . 2 (((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ∧ 𝑁𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒) ↔ ((𝐾 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ (Vtx‘𝐺)) ∧ 𝐾𝑁 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒))
7 eqid 2824 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
8 eqid 2824 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
97, 8nbgrel 27119 . 2 (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ ((𝑁 ∈ (Vtx‘𝐺) ∧ 𝐾 ∈ (Vtx‘𝐺)) ∧ 𝑁𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒))
107, 8nbgrel 27119 . 2 (𝐾 ∈ (𝐺 NeighbVtx 𝑁) ↔ ((𝐾 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ (Vtx‘𝐺)) ∧ 𝐾𝑁 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝐾} ⊆ 𝑒))
116, 9, 103bitr4i 306 1 (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 NeighbVtx 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084  wcel 2115  wne 3013  wrex 3133  wss 3918  {cpr 4550  cfv 6336  (class class class)co 7138  Vtxcvtx 26778  Edgcedg 26829   NeighbVtx cnbgr 27111
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-1st 7672  df-2nd 7673  df-nbgr 27112
This theorem is referenced by:  nbusgredgeu0  27147  uvtxnbgrvtx  27172  cplgr3v  27214  frgrncvvdeqlem1  28073  frgrwopreglem4a  28084  numclwwlk1lem2foa  28128
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