Theorem dfclnbgr3 48619

Description: Alternate definition of the closed neighborhood of a vertex using the edge function instead of the edges themselves (see also clnbgrval 48615). (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 29408	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	2	eqcomi 2772	. . . 4 ⊢ ran (iEdg‘𝐺) = (Edg‘𝐺)
4	1, 3	clnbgrval 48615	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ ran (iEdg‘𝐺){𝑁, 𝑛} ⊆ 𝑒}))
5	4	adantr 485	. 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 2772	. . . . . . 7 ⊢ (iEdg‘𝐺) = 𝐼
8	7	rneqi 5927	. . . . . 6 ⊢ ran (iEdg‘𝐺) = ran 𝐼
9	8	rexeqi 3322	. . . . 5 ⊢ (∃𝑒 ∈ ran (iEdg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒)
10		funfn 6566	. . . . . . 7 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼)
11	10	bilani 509	. . . . . 6 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → 𝐼 Fn dom 𝐼)
12		sseq2 3963	. . . . . . 7 ⊢ (𝑒 = (𝐼‘𝑖) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
13	12	rexrn 7082	. . . . . 6 ⊢ (𝐼 Fn dom 𝐼 → (∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
14	11, 13	syl 18	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
15	9, 14	bitrid 286	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ ran (iEdg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
16	15	rabbidv 3423	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ ran (iEdg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ 𝑉 ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)})
17	16	uneq2d 4122	. 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)}))
18	5, 17	eqtrd 2798	1 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  {crab 3416   ∪ cun 3903   ⊆ wss 3905  {csn 4589  {cpr 4591  dom cdm 5661  ran crn 5662  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410  Vtxcvtx 29355  iEdgciedg 29356  Edgcedg 29406   ClNeighbVtx cclnbgr 48611

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-edg 29407  df-clnbgr 48612

This theorem is used by: (None)