Theorem vopnbgrelself 47482

Description: A vertex 𝑁 is a member of its semiopen neighborhood iff there is a loop joining the vertex with itself. (Contributed by AV, 16-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
vopnbgrelself	⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑈 ↔ ∃𝑒 ∈ 𝐸 𝑒 = {𝑁}))

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

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

Proof of Theorem vopnbgrelself

Step	Hyp	Ref	Expression
1		ibar 527	. 2 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})) ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})))))
2		eqid 2729	. . . . . . 7 ⊢ 𝑁 = 𝑁
3	2	jctl 522	. . . . . 6 ⊢ (𝑒 = {𝑁} → (𝑁 = 𝑁 ∧ 𝑒 = {𝑁}))
4	3	olcd 872	. . . . 5 ⊢ (𝑒 = {𝑁} → ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})))
5		eqneqall 2944	. . . . . . . 8 ⊢ (𝑁 = 𝑁 → (𝑁 ≠ 𝑁 → ((𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) → 𝑒 = {𝑁})))
6	2, 5	ax-mp 5	. . . . . . 7 ⊢ (𝑁 ≠ 𝑁 → ((𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) → 𝑒 = {𝑁}))
7	6	3impib 1113	. . . . . 6 ⊢ ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) → 𝑒 = {𝑁})
8		simpr 483	. . . . . 6 ⊢ ((𝑁 = 𝑁 ∧ 𝑒 = {𝑁}) → 𝑒 = {𝑁})
9	7, 8	jaoi 855	. . . . 5 ⊢ (((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})) → 𝑒 = {𝑁})
10	4, 9	impbii 208	. . . 4 ⊢ (𝑒 = {𝑁} ↔ ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})))
11	10	a1i 11	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑒 = {𝑁} ↔ ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁}))))
12	11	rexbidv 3172	. 2 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 𝑒 = {𝑁} ↔ ∃𝑒 ∈ 𝐸 ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁}))))
13		dfvopnbgr2.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
14		dfvopnbgr2.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
15		dfvopnbgr2.u	. . 3 ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}
16	13, 14, 15	vopnbgrel 47481	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑈 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})))))
17	1, 12, 16	3bitr4rd 311	1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑈 ↔ ∃𝑒 ∈ 𝐸 𝑒 = {𝑁}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1534   ∈ wcel 2100   ≠ wne 2933  ∃wrex 3063  {crab 3428  {csn 4634  ‘cfv 6556  (class class class)co 7428  Vtxcvtx 28955  Edgcedg 29006   NeighbVtx cnbgr 29291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-sep 5305  ax-nul 5312  ax-pr 5435  ax-un 7749

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-sbc 3787  df-csb 3903  df-dif 3960  df-un 3962  df-in 3964  df-ss 3974  df-nul 4334  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4917  df-iun 5006  df-br 5155  df-opab 5217  df-mpt 5238  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fv 6564  df-ov 7431  df-oprab 7432  df-mpo 7433  df-1st 8009  df-2nd 8010  df-nbgr 29292

This theorem is referenced by: (None)