Theorem dfvopnbgr2 47952

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 29318	. . . . . . 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 916	. . . . 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 1088	. . . . . . . . 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 3164	. . . . . . . . 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 278	. . . . . . . 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 913	. . . . . . 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 3100	. . . . . 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 289	. . . . 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 279	. . . 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 468	. . . . . . . 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 528	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → ((𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ (𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒))))
18	17	ancoms 458	. . . . . . . 8 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ (𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒))))
19	18	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ((𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ (𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒))))
20		prssg 4768	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
21	20	bicomd 223	. . . . . . . . . 10 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
22	21	adantr 480	. . . . . . . . 9 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
23	22	anbi2d 630	. . . . . . . 8 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ((𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ (𝑛 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒))))
24		3anass 1094	. . . . . . . 8 ⊢ ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (𝑛 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
25	23, 24	bitr4di 289	. . . . . . 7 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ((𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ (𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
26	16, 19, 25	3bitr2d 307	. . . . . 6 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ((((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ (𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
27		eqcom 2738	. . . . . . . . 9 ⊢ (𝑁 = 𝑛 ↔ 𝑛 = 𝑁)
28	27	anbi1i 624	. . . . . . . 8 ⊢ ((𝑁 = 𝑛 ∧ 𝑒 = {𝑁}) ↔ (𝑛 = 𝑁 ∧ 𝑒 = {𝑁}))
29		sneq 4583	. . . . . . . . . . 11 ⊢ (𝑁 = 𝑛 → {𝑁} = {𝑛})
30	29	eqcoms 2739	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → {𝑁} = {𝑛})
31	30	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑒 = {𝑁} ↔ 𝑒 = {𝑛}))
32	31	pm5.32i 574	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑒 = {𝑁}) ↔ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))
33	28, 32	bitri 275	. . . . . . 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 918	. . . . 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 3154	. . . 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 279	. . 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 3401	. 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 2778	1 ⊢ (𝑁 ∈ 𝑉 → 𝑈 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056  {crab 3395   ⊆ wss 3897  {csn 4573  {cpr 4575  ‘cfv 6481  (class class class)co 7346  Vtxcvtx 28974  Edgcedg 29025   NeighbVtx cnbgr 29310

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-sbc 3737  df-csb 3846  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-iun 4941  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-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-nbgr 29311

This theorem is referenced by:  vopnbgrel  47953  dfclnbgr6  47955  dfnbgr6  47956  dfsclnbgr6  47957