Theorem dfvopnbgr2 48480

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 29543	. . . . . . 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 927	. . . . 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 1101	. . . . . . . . 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 3196	. . . . . . . . 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 280	. . . . . . . 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 924	. . . . . . 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 3132	. . . . . 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 291	. . . . 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 281	. . . 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 472	. . . . . . . 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 536	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → ((𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ (𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒))))
18	17	ancoms 462	. . . . . . . 8 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ (𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒))))
19	18	adantr 484	. . . . . . 7 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ((𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ (𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒))))
20		prssg 4779	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
21	20	bicomd 225	. . . . . . . . . 10 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
22	21	adantr 484	. . . . . . . . 9 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
23	22	anbi2d 639	. . . . . . . 8 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ((𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ (𝑛 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒))))
24		3anass 1107	. . . . . . . 8 ⊢ ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (𝑛 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
25	23, 24	bitr4di 291	. . . . . . 7 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ((𝑛 ≠ 𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ (𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
26	16, 19, 25	3bitr2d 309	. . . . . 6 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ∧ 𝑒 ∈ 𝐸) → ((((𝑛 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑛 ≠ 𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ (𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
27		eqcom 2771	. . . . . . . . 9 ⊢ (𝑁 = 𝑛 ↔ 𝑛 = 𝑁)
28	27	anbi1i 633	. . . . . . . 8 ⊢ ((𝑁 = 𝑛 ∧ 𝑒 = {𝑁}) ↔ (𝑛 = 𝑁 ∧ 𝑒 = {𝑁}))
29		sneq 4594	. . . . . . . . . . 11 ⊢ (𝑁 = 𝑛 → {𝑁} = {𝑛})
30	29	eqcoms 2772	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → {𝑁} = {𝑛})
31	30	eqeq2d 2775	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑒 = {𝑁} ↔ 𝑒 = {𝑛}))
32	31	pm5.32i 582	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑒 = {𝑁}) ↔ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))
33	28, 32	bitri 277	. . . . . . 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 929	. . . . 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 3186	. . . 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 281	. . 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 3422	. 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 2811	1 ⊢ (𝑁 ∈ 𝑉 → 𝑈 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  ∃wrex 3088  {crab 3416   ⊆ wss 3906  {csn 4584  {cpr 4586  ‘cfv 6523  (class class class)co 7398  Vtxcvtx 29199  Edgcedg 29250   NeighbVtx cnbgr 29535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-nbgr 29536

This theorem is referenced by:  vopnbgrel  48481  dfclnbgr6  48483  dfnbgr6  48484  dfsclnbgr6  48485