Theorem dfvopnbgr2 48541

Description: Alternate definition of the semiopen neighborhood of a vertex breaking up the subset relationship of an unordered pair. A semiopen neighborhood 𝑈 of a vertex 𝑁 is its open neighborhood together with itself if there is a loop at this vertex. (Contributed by AV, 15-May-2025.)

Hypotheses

Ref	Expression
dfvopnbgr2.v	⊢ 𝑉 = (Vtx‘𝐺)
dfvopnbgr2.e	⊢ 𝐸 = (Edg‘𝐺)
dfvopnbgr2.u	⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}

Assertion

Ref	Expression
dfvopnbgr2	⊢ (𝑁 ∈ 𝑉 → 𝑈 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})

Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛

Allowed substitution hints:   𝑈(𝑒,𝑛)   𝐸(𝑛)   𝐺(𝑛)

Proof of Theorem dfvopnbgr2

Step	Hyp	Ref	Expression
1		dfvopnbgr2.u	. 2 ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}
2		dfvopnbgr2.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
3		dfvopnbgr2.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
4	2, 3	nbgrel 29631	. . . . . . 7 ⊢ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ↔ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
5	4	a1i 11	. . . . . 6 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ↔ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒)))
6	5	orbi1d 929	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁})) ↔ (((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))))
7		df-3an 1103	. . . . . . . . 9 ⊢ (((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒) ↔ (((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
8		r19.42v 3203	. . . . . . . . 9 ⊢ (∃𝑒 ∈ 𝐸 (((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ (((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
9	7, 8	bitr4i 281	. . . . . . . 8 ⊢ (((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒) ↔ ∃𝑒 ∈ 𝐸 (((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒))
10	9	orbi1i 926	. . . . . . 7 ⊢ ((((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁})) ↔ (∃𝑒 ∈ 𝐸 (((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁})))
11	10	a1i 11	. . . . . 6 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁})) ↔ (∃𝑒 ∈ 𝐸 (((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))))
12		r19.43 3139	. . . . . 6 ⊢ (∃𝑒 ∈ 𝐸 ((((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ (𝑁 = 𝑛 ∧ 𝑒 = {𝑁})) ↔ (∃𝑒 ∈ 𝐸 (((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁})))
13	11, 12	bitr4di 292	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁})) ↔ ∃𝑒 ∈ 𝐸 ((((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))))
14	6, 13	bitrd 282	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁})) ↔ ∃𝑒 ∈ 𝐸 ((((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))))
15		anass 473	. . . . . . . 8 ⊢ ((((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ (𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒)))
16	15	a1i 11	. . . . . . 7 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ((((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ (𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒))))
17		ibar 537	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → ((𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ (𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒))))
18	17	ancoms 463	. . . . . . . 8 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ (𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒))))
19	18	adantr 485	. . . . . . 7 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ((𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ (𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒))))
20		prssg 4789	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
21	20	bicomd 226	. . . . . . . . . 10 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
22	21	adantr 485	. . . . . . . . 9 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
23	22	anbi2d 641	. . . . . . . 8 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ((𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ (𝑛 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒))))
24		3anass 1109	. . . . . . . 8 ⊢ ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (𝑛 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
25	23, 24	bitr4di 292	. . . . . . 7 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ((𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ (𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
26	16, 19, 25	3bitr2d 310	. . . . . 6 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ((((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ (𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
27		eqcom 2776	. . . . . . . . 9 ⊢ (𝑁 = 𝑛 ↔ 𝑛 = 𝑁)
28	27	anbi1i 635	. . . . . . . 8 ⊢ ((𝑁 = 𝑛 ∧ 𝑒 = {𝑁}) ↔ (𝑛 = 𝑁 ∧ 𝑒 = {𝑁}))
29		sneq 4604	. . . . . . . . . . 11 ⊢ (𝑁 = 𝑛 → {𝑁} = {𝑛})
30	29	eqcoms 2777	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → {𝑁} = {𝑛})
31	30	eqeq2d 2780	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑒 = {𝑁} ↔ 𝑒 = {𝑛}))
32	31	pm5.32i 584	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑒 = {𝑁}) ↔ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))
33	28, 32	bitri 278	. . . . . . 7 ⊢ ((𝑁 = 𝑛 ∧ 𝑒 = {𝑁}) ↔ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))
34	33	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ((𝑁 = 𝑛 ∧ 𝑒 = {𝑁}) ↔ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})))
35	26, 34	orbi12d 931	. . . . 5 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → (((((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ (𝑁 = 𝑛 ∧ 𝑒 = {𝑁})) ↔ ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))))
36	35	rexbidva 3193	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → (∃𝑒 ∈ 𝐸 ((((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ (𝑁 = 𝑛 ∧ 𝑒 = {𝑁})) ↔ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))))
37	14, 36	bitrd 282	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁})) ↔ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))))
38	37	rabbidva 3429	. 2 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))} = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})
39	1, 38	eqtrid 2816	1 ⊢ (𝑁 ∈ 𝑉 → 𝑈 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095  {crab 3423   ⊆ wss 3913  {csn 4594  {cpr 4596  ‘cfv 6537  (class class class)co 7411  Vtxcvtx 29287  Edgcedg 29338   NeighbVtx cnbgr 29623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-nbgr 29624

This theorem is referenced by:  vopnbgrel  48542  dfclnbgr6  48544  dfnbgr6  48545  dfsclnbgr6  48546