Theorem dfclnbgr4 48316

Description: Alternate definition of the closed neighborhood of a vertex as union of the vertex with its open neighborhood. (Contributed by AV, 8-May-2025.)

Hypothesis

Ref	Expression
dfclnbgr4.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
dfclnbgr4	⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))

Proof of Theorem dfclnbgr4

Dummy variables 𝑒 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfclnbgr4.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		eqid 2737	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	dfclnbgr2 48315	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}))
4		undif2 4418	. . . 4 ⊢ ({𝑁} ∪ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁})) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
5		rabdif 4262	. . . . 5 ⊢ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁}) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}
6	5	uneq2i 4106	. . . 4 ⊢ ({𝑁} ∪ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁})) = ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
7	4, 6	eqtr3i 2762	. . 3 ⊢ ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) = ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
8	1, 2	dfnbgr2 29424	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
9	8	eqcomd 2743	. . . 4 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} = (𝐺 NeighbVtx 𝑁))
10	9	uneq2d 4109	. . 3 ⊢ (𝑁 ∈ 𝑉 → ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
11	7, 10	eqtrid 2784	. 2 ⊢ (𝑁 ∈ 𝑉 → ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
12	3, 11	eqtrd 2772	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3390   ∖ cdif 3887   ∪ cun 3888  {csn 4568  ‘cfv 6494  (class class class)co 7362  Vtxcvtx 29083  Edgcedg 29134   NeighbVtx cnbgr 29419   ClNeighbVtx cclnbgr 48310

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-iota 6450  df-fun 6496  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-nbgr 29420  df-clnbgr 48311

This theorem is referenced by:  elclnbgrelnbgr  48317  clnbupgr  48325  clnbgr0edg  48329  edgusgrclnbfin  48334  stgrclnbgr0  48457  isubgr3stgrlem1  48458