Theorem vopnbgrel 47838

Description: Characterization of a member 𝑋 of the semiopen neighborhood of a vertex 𝑁 in a graph 𝐺. (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
vopnbgrel	⊢ (𝑁 ∈ 𝑉 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋})))))

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

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

Proof of Theorem vopnbgrel

Step	Hyp	Ref	Expression
1		dfvopnbgr2.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		dfvopnbgr2.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
3		dfvopnbgr2.u	. . . 4 ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}
4	1, 2, 3	dfvopnbgr2 47837	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝑈 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})
5	4	eleq2d 2814	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))}))
6		neeq1 2987	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑛 ≠ 𝑁 ↔ 𝑋 ≠ 𝑁))
7		eleq1 2816	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑛 ∈ 𝑒 ↔ 𝑋 ∈ 𝑒))
8	6, 7	3anbi13d 1440	. . . . 5 ⊢ (𝑛 = 𝑋 → ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒)))
9		eqeq1 2733	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑛 = 𝑁 ↔ 𝑋 = 𝑁))
10		sneq 4587	. . . . . . 7 ⊢ (𝑛 = 𝑋 → {𝑛} = {𝑋})
11	10	eqeq2d 2740	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑒 = {𝑛} ↔ 𝑒 = {𝑋}))
12	9, 11	anbi12d 632	. . . . 5 ⊢ (𝑛 = 𝑋 → ((𝑛 = 𝑁 ∧ 𝑒 = {𝑛}) ↔ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋})))
13	8, 12	orbi12d 918	. . . 4 ⊢ (𝑛 = 𝑋 → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ↔ ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋}))))
14	13	rexbidv 3153	. . 3 ⊢ (𝑛 = 𝑋 → (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ↔ ∃𝑒 ∈ 𝐸 ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋}))))
15	14	elrab 3648	. 2 ⊢ (𝑋 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋}))))
16	5, 15	bitrdi 287	1 ⊢ (𝑁 ∈ 𝑉 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋})))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  {crab 3394  {csn 4577  ‘cfv 6482  (class class class)co 7349  Vtxcvtx 28941  Edgcedg 28992   NeighbVtx cnbgr 29277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-nbgr 29278

This theorem is referenced by:  vopnbgrelself  47839