MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nbgrval Structured version   Visualization version   GIF version

Theorem nbgrval 26522
Description: The set of neighbors of a vertex 𝑉 in a graph 𝐺. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 24-Oct-2020.) (Revised by AV, 21-Mar-2021.)
Hypotheses
Ref Expression
nbgrval.v 𝑉 = (Vtx‘𝐺)
nbgrval.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
nbgrval (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺,𝑛   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛
Allowed substitution hint:   𝐸(𝑛)

Proof of Theorem nbgrval
Dummy variables 𝑔 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nbgr 26518 . 2 NeighbVtx = (𝑔 ∈ V, 𝑘 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒})
2 nbgrval.v . . . 4 𝑉 = (Vtx‘𝐺)
321vgrex 26185 . . 3 (𝑁𝑉𝐺 ∈ V)
4 fveq2 6379 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
54, 2syl6reqr 2818 . . . . 5 (𝑔 = 𝐺𝑉 = (Vtx‘𝑔))
65eleq2d 2830 . . . 4 (𝑔 = 𝐺 → (𝑁𝑉𝑁 ∈ (Vtx‘𝑔)))
76biimpac 470 . . 3 ((𝑁𝑉𝑔 = 𝐺) → 𝑁 ∈ (Vtx‘𝑔))
8 fvex 6392 . . . . 5 (Vtx‘𝑔) ∈ V
98difexi 4972 . . . 4 ((Vtx‘𝑔) ∖ {𝑘}) ∈ V
10 rabexg 4974 . . . 4 (((Vtx‘𝑔) ∖ {𝑘}) ∈ V → {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒} ∈ V)
119, 10mp1i 13 . . 3 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒} ∈ V)
124, 2syl6eqr 2817 . . . . . . 7 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
1312adantr 472 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → (Vtx‘𝑔) = 𝑉)
14 sneq 4346 . . . . . . 7 (𝑘 = 𝑁 → {𝑘} = {𝑁})
1514adantl 473 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → {𝑘} = {𝑁})
1613, 15difeq12d 3893 . . . . 5 ((𝑔 = 𝐺𝑘 = 𝑁) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
1716adantl 473 . . . 4 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
18 fveq2 6379 . . . . . . . 8 (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
19 nbgrval.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
2018, 19syl6eqr 2817 . . . . . . 7 (𝑔 = 𝐺 → (Edg‘𝑔) = 𝐸)
2120adantr 472 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → (Edg‘𝑔) = 𝐸)
2221adantl 473 . . . . 5 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → (Edg‘𝑔) = 𝐸)
23 preq1 4425 . . . . . . . 8 (𝑘 = 𝑁 → {𝑘, 𝑛} = {𝑁, 𝑛})
2423sseq1d 3794 . . . . . . 7 (𝑘 = 𝑁 → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
2524adantl 473 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
2625adantl 473 . . . . 5 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
2722, 26rexeqbidv 3301 . . . 4 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → (∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
2817, 27rabeqbidv 3344 . . 3 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
293, 7, 11, 28ovmpt2dv2 6996 . 2 (𝑁𝑉 → ( NeighbVtx = (𝑔 ∈ V, 𝑘 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒}) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
301, 29mpi 20 1 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wrex 3056  {crab 3059  Vcvv 3350  cdif 3731  wss 3734  {csn 4336  {cpr 4338  cfv 6070  (class class class)co 6846  cmpt2 6848  Vtxcvtx 26179  Edgcedg 26230   NeighbVtx cnbgr 26517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-iota 6033  df-fun 6072  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-nbgr 26518
This theorem is referenced by:  dfnbgr2  26523  dfnbgr3  26524  nbgrel  26526  nbgrelOLD  26527  nbuhgr  26532  nbupgr  26533  nbumgrvtx  26535  nbgr0vtxlem  26544  nbgrnself  26548
  Copyright terms: Public domain W3C validator