Theorem dfclnbgr3 47813

Description: Alternate definition of the closed neighborhood of a vertex using the edge function instead of the edges themselves (see also clnbgrval 47809). (Contributed by AV, 8-May-2025.)

Hypotheses

Ref	Expression
dfclnbgr3.v	⊢ 𝑉 = (Vtx‘𝐺)
dfclnbgr3.i	⊢ 𝐼 = (iEdg‘𝐺)

Assertion

Ref	Expression
dfclnbgr3	⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)}))

Distinct variable groups:   𝑛,𝐺   𝑖,𝐼,𝑛   𝑖,𝑁,𝑛   𝑛,𝑉

Allowed substitution hints:   𝐺(𝑖)   𝑉(𝑖)

Proof of Theorem dfclnbgr3

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfclnbgr3.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		edgval 29066	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	2	eqcomi 2746	. . . 4 ⊢ ran (iEdg‘𝐺) = (Edg‘𝐺)
4	1, 3	clnbgrval 47809	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ ran (iEdg‘𝐺){𝑁, 𝑛} ⊆ 𝑒}))
5	4	adantr 480	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ ran (iEdg‘𝐺){𝑁, 𝑛} ⊆ 𝑒}))
6		dfclnbgr3.i	. . . . . . . 8 ⊢ 𝐼 = (iEdg‘𝐺)
7	6	eqcomi 2746	. . . . . . 7 ⊢ (iEdg‘𝐺) = 𝐼
8	7	rneqi 5948	. . . . . 6 ⊢ ran (iEdg‘𝐺) = ran 𝐼
9	8	rexeqi 3325	. . . . 5 ⊢ (∃𝑒 ∈ ran (iEdg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒)
10		funfn 6596	. . . . . . . 8 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼)
11	10	biimpi 216	. . . . . . 7 ⊢ (Fun 𝐼 → 𝐼 Fn dom 𝐼)
12	11	adantl 481	. . . . . 6 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → 𝐼 Fn dom 𝐼)
13		sseq2 4010	. . . . . . 7 ⊢ (𝑒 = (𝐼‘𝑖) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
14	13	rexrn 7107	. . . . . 6 ⊢ (𝐼 Fn dom 𝐼 → (∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
15	12, 14	syl 17	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
16	9, 15	bitrid 283	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ ran (iEdg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
17	16	rabbidv 3444	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ ran (iEdg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ 𝑉 ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)})
18	17	uneq2d 4168	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ ran (iEdg‘𝐺){𝑁, 𝑛} ⊆ 𝑒}) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)}))
19	5, 18	eqtrd 2777	1 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  {crab 3436   ∪ cun 3949   ⊆ wss 3951  {csn 4626  {cpr 4628  dom cdm 5685  ran crn 5686  Fun wfun 6555   Fn wfn 6556  ‘cfv 6561  (class class class)co 7431  Vtxcvtx 29013  iEdgciedg 29014  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-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-edg 29065  df-clnbgr 47806

This theorem is referenced by: (None)