Theorem dfclnbgr3 47751

Description: Alternate definition of the closed neighborhood of a vertex using the edge function instead of the edges themselves (see also clnbgrval 47747). (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 29081	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	2	eqcomi 2744	. . . 4 ⊢ ran (iEdg‘𝐺) = (Edg‘𝐺)
4	1, 3	clnbgrval 47747	. . 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 2744	. . . . . . 7 ⊢ (iEdg‘𝐺) = 𝐼
8	7	rneqi 5951	. . . . . 6 ⊢ ran (iEdg‘𝐺) = ran 𝐼
9	8	rexeqi 3323	. . . . 5 ⊢ (∃𝑒 ∈ ran (iEdg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒)
10		funfn 6598	. . . . . . . 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 4022	. . . . . . 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 3441	. . 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 4178	. 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 2775	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 1537   ∈ wcel 2106  ∃wrex 3068  {crab 3433   ∪ cun 3961   ⊆ wss 3963  {csn 4631  {cpr 4633  dom cdm 5689  ran crn 5690  Fun wfun 6557   Fn wfn 6558  ‘cfv 6563  (class class class)co 7431  Vtxcvtx 29028  iEdgciedg 29029  Edgcedg 29079   ClNeighbVtx cclnbgr 47743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-edg 29080  df-clnbgr 47744

This theorem is referenced by: (None)