Theorem dfclnbgr3 48072

Description: Alternate definition of the closed neighborhood of a vertex using the edge function instead of the edges themselves (see also clnbgrval 48068). (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 29122	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	2	eqcomi 2745	. . . 4 ⊢ ran (iEdg‘𝐺) = (Edg‘𝐺)
4	1, 3	clnbgrval 48068	. . 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 2745	. . . . . . 7 ⊢ (iEdg‘𝐺) = 𝐼
8	7	rneqi 5886	. . . . . 6 ⊢ ran (iEdg‘𝐺) = ran 𝐼
9	8	rexeqi 3295	. . . . 5 ⊢ (∃𝑒 ∈ ran (iEdg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒)
10		funfn 6522	. . . . . . . 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 3960	. . . . . . 7 ⊢ (𝑒 = (𝐼‘𝑖) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
14	13	rexrn 7032	. . . . . 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 3406	. . 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 4120	. 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 2771	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 2113  ∃wrex 3060  {crab 3399   ∪ cun 3899   ⊆ wss 3901  {csn 4580  {cpr 4582  dom cdm 5624  ran crn 5625  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492  (class class class)co 7358  Vtxcvtx 29069  iEdgciedg 29070  Edgcedg 29120   ClNeighbVtx cclnbgr 48064

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-edg 29121  df-clnbgr 48065

This theorem is referenced by: (None)