Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clnbgrval Structured version   Visualization version   GIF version

Theorem clnbgrval 47809
Description: The closed neighborhood of a vertex 𝑉 in a graph 𝐺. (Contributed by AV, 7-May-2025.)
Hypotheses
Ref Expression
clnbgrval.v 𝑉 = (Vtx‘𝐺)
clnbgrval.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
clnbgrval (𝑁𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛𝑉 ∣ ∃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 clnbgrval
Dummy variables 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-clnbgr 47806 . 2 ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}))
2 clnbgrval.v . . . 4 𝑉 = (Vtx‘𝐺)
321vgrex 29019 . . 3 (𝑁𝑉𝐺 ∈ V)
4 fveq2 6906 . . . . . . 7 (𝐺 = 𝑔 → (Vtx‘𝐺) = (Vtx‘𝑔))
54eqcoms 2745 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝐺) = (Vtx‘𝑔))
62, 5eqtrid 2789 . . . . 5 (𝑔 = 𝐺𝑉 = (Vtx‘𝑔))
76eleq2d 2827 . . . 4 (𝑔 = 𝐺 → (𝑁𝑉𝑁 ∈ (Vtx‘𝑔)))
87biimpac 478 . . 3 ((𝑁𝑉𝑔 = 𝐺) → 𝑁 ∈ (Vtx‘𝑔))
9 vsnex 5434 . . . . 5 {𝑣} ∈ V
109a1i 11 . . . 4 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑣 = 𝑁)) → {𝑣} ∈ V)
11 fvex 6919 . . . . 5 (Vtx‘𝑔) ∈ V
12 rabexg 5337 . . . . 5 ((Vtx‘𝑔) ∈ V → {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒} ∈ V)
1311, 12mp1i 13 . . . 4 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑣 = 𝑁)) → {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒} ∈ V)
1410, 13unexd 7774 . . 3 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑣 = 𝑁)) → ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) ∈ V)
15 sneq 4636 . . . . . 6 (𝑣 = 𝑁 → {𝑣} = {𝑁})
1615adantl 481 . . . . 5 ((𝑔 = 𝐺𝑣 = 𝑁) → {𝑣} = {𝑁})
17 fveq2 6906 . . . . . . . 8 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
1817, 2eqtr4di 2795 . . . . . . 7 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
1918adantr 480 . . . . . 6 ((𝑔 = 𝐺𝑣 = 𝑁) → (Vtx‘𝑔) = 𝑉)
20 fveq2 6906 . . . . . . . . 9 (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
21 clnbgrval.e . . . . . . . . 9 𝐸 = (Edg‘𝐺)
2220, 21eqtr4di 2795 . . . . . . . 8 (𝑔 = 𝐺 → (Edg‘𝑔) = 𝐸)
2322adantr 480 . . . . . . 7 ((𝑔 = 𝐺𝑣 = 𝑁) → (Edg‘𝑔) = 𝐸)
24 preq1 4733 . . . . . . . . 9 (𝑣 = 𝑁 → {𝑣, 𝑛} = {𝑁, 𝑛})
2524sseq1d 4015 . . . . . . . 8 (𝑣 = 𝑁 → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
2625adantl 481 . . . . . . 7 ((𝑔 = 𝐺𝑣 = 𝑁) → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
2723, 26rexeqbidv 3347 . . . . . 6 ((𝑔 = 𝐺𝑣 = 𝑁) → (∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
2819, 27rabeqbidv 3455 . . . . 5 ((𝑔 = 𝐺𝑣 = 𝑁) → {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒} = {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
2916, 28uneq12d 4169 . . . 4 ((𝑔 = 𝐺𝑣 = 𝑁) → ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) = ({𝑁} ∪ {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
3029adantl 481 . . 3 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑣 = 𝑁)) → ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) = ({𝑁} ∪ {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
313, 8, 14, 30ovmpodv2 7591 . 2 (𝑁𝑉 → ( ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})))
321, 31mpi 20 1 (𝑁𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  {crab 3436  Vcvv 3480  cun 3949  wss 3951  {csn 4626  {cpr 4628  cfv 6561  (class class class)co 7431  cmpo 7433  Vtxcvtx 29013  Edgcedg 29064   ClNeighbVtx cclnbgr 47805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-clnbgr 47806
This theorem is referenced by:  dfclnbgr2  47810  dfclnbgr3  47813  clnbgrel  47815
  Copyright terms: Public domain W3C validator