Theorem dfclnbgr3 47936

Description: Alternate definition of the closed neighborhood of a vertex using the edge function instead of the edges themselves (see also clnbgrval 47932). (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 29027	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	2	eqcomi 2740	. . . 4 ⊢ ran (iEdg‘𝐺) = (Edg‘𝐺)
4	1, 3	clnbgrval 47932	. . 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 2740	. . . . . . 7 ⊢ (iEdg‘𝐺) = 𝐼
8	7	rneqi 5876	. . . . . 6 ⊢ ran (iEdg‘𝐺) = ran 𝐼
9	8	rexeqi 3291	. . . . 5 ⊢ (∃𝑒 ∈ ran (iEdg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒)
10		funfn 6511	. . . . . . . 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 3956	. . . . . . 7 ⊢ (𝑒 = (𝐼‘𝑖) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
14	13	rexrn 7020	. . . . . 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 3402	. . 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 4115	. 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 2766	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 1541   ∈ wcel 2111  ∃wrex 3056  {crab 3395   ∪ cun 3895   ⊆ wss 3897  {csn 4573  {cpr 4575  dom cdm 5614  ran crn 5615  Fun wfun 6475   Fn wfn 6476  ‘cfv 6481  (class class class)co 7346  Vtxcvtx 28974  iEdgciedg 28975  Edgcedg 29025   ClNeighbVtx cclnbgr 47928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-edg 29026  df-clnbgr 47929

This theorem is referenced by: (None)