Theorem vopnbgrelself 48477

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 536	. 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 2762	. . . . . . 7 ⊢ 𝑁 = 𝑁
3	2	jctl 531	. . . . . 6 ⊢ (𝑒 = {𝑁} → (𝑁 = 𝑁 ∧ 𝑒 = {𝑁}))
4	3	olcd 885	. . . . 5 ⊢ (𝑒 = {𝑁} → ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})))
5		eqneqall 2968	. . . . . . . 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 1129	. . . . . 6 ⊢ ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) → 𝑒 = {𝑁})
8		simpr 488	. . . . . 6 ⊢ ((𝑁 = 𝑁 ∧ 𝑒 = {𝑁}) → 𝑒 = {𝑁})
9	7, 8	jaoi 868	. . . . 5 ⊢ (((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})) → 𝑒 = {𝑁})
10	4, 9	impbii 211	. . . 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 3186	. 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 48476	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑈 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})))))
17	1, 12, 16	3bitr4rd 314	1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑈 ↔ ∃𝑒 ∈ 𝐸 𝑒 = {𝑁}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3086  {crab 3414  {csn 4582  ‘cfv 6521  (class class class)co 7396  Vtxcvtx 29197  Edgcedg 29248   NeighbVtx cnbgr 29533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-nbgr 29534

This theorem is referenced by: (None)