Theorem vopnbgrelself 48508

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 537	. 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 2769	. . . . . . 7 ⊢ 𝑁 = 𝑁
3	2	jctl 532	. . . . . 6 ⊢ (𝑒 = {𝑁} → (𝑁 = 𝑁 ∧ 𝑒 = {𝑁}))
4	3	olcd 887	. . . . 5 ⊢ (𝑒 = {𝑁} → ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})))
5		eqneqall 2975	. . . . . . . 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 1132	. . . . . 6 ⊢ ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) → 𝑒 = {𝑁})
8		simpr 489	. . . . . 6 ⊢ ((𝑁 = 𝑁 ∧ 𝑒 = {𝑁}) → 𝑒 = {𝑁})
9	7, 8	jaoi 870	. . . . 5 ⊢ (((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})) → 𝑒 = {𝑁})
10	4, 9	impbii 212	. . . 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 3195	. 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 48507	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑈 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})))))
17	1, 12, 16	3bitr4rd 315	1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑈 ↔ ∃𝑒 ∈ 𝐸 𝑒 = {𝑁}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095  {crab 3423  {csn 4594  ‘cfv 6537  (class class class)co 7411  Vtxcvtx 29286  Edgcedg 29337   NeighbVtx cnbgr 29622

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 7985  df-2nd 7986  df-nbgr 29623

This theorem is referenced by: (None)