Theorem nbgrel 29800

Description: Characterization of a neighbor 𝑁 of a vertex 𝑋 in a graph 𝐺. (Contributed by Alexander van der Vekens and Mario Carneiro, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Revised by AV, 12-Feb-2022.)

Hypotheses

Ref	Expression
nbgrel.v	⊢ 𝑉 = (Vtx‘𝐺)
nbgrel.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
nbgrel	⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))

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

Proof of Theorem nbgrel

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbgrel.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	nbgrcl 29795	. . 3 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ 𝑉)
3	2	pm4.71ri 570	. 2 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝑋)))
4		nbgrel.e	. . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺)
5	1, 4	nbgrval 29796	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝐺 NeighbVtx 𝑋) = {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒})
6	5	eleq2d 2846	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}))
7		preq2 4695	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → {𝑋, 𝑛} = {𝑋, 𝑁})
8	7	sseq1d 3962	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ({𝑋, 𝑛} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ 𝑒))
9	8	rexbidv 3186	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
10	9	elrab 3645	. . . . . 6 ⊢ (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ (𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
11		eldifsn 4748	. . . . . . 7 ⊢ (𝑁 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋))
12	11	anbi1i 636	. . . . . 6 ⊢ ((𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
13	10, 12	bitri 278	. . . . 5 ⊢ (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
14	6, 13	bitrdi 290	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
15	14	pm5.32i 585	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ (𝑋 ∈ 𝑉 ∧ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
16		df-3an 1105	. . . 4 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
17		anass 474	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋) ↔ (𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋)))
18		ancom 466	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ↔ (𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉))
19	18	anbi1i 636	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋))
20	17, 19	bitr3i 280	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋)) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋))
21	20	anbi1i 636	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋)) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
22		anass 474	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋)) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ (𝑋 ∈ 𝑉 ∧ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
23	16, 21, 22	3bitr2ri 303	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
24	15, 23	bitri 278	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
25	3, 24	bitri 278	1 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086  {crab 3412   ∖ cdif 3896   ⊆ wss 3899  {csn 4584  {cpr 4586  ‘cfv 6533  (class class class)co 7413  Vtxcvtx 29453  Edgcedg 29504   NeighbVtx cnbgr 29792

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-nbgr 29793

This theorem is used by:  nbgrisvtx  29801  nbgr2vtx1edg  29810  nbuhgr2vtx1edgblem  29811  nbuhgr2vtx1edgb  29812  nbgrsym  29823  isuvtx  29855  iscplgredg  29877  cusgrexi  29903  structtocusgr  29906  dfvopnbgr2  48769