Theorem nbgrel 29427

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 29422	. . 3 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ 𝑉)
3	2	pm4.71ri 565	. 2 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝑋)))
4		nbgrel.e	. . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺)
5	1, 4	nbgrval 29423	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝐺 NeighbVtx 𝑋) = {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒})
6	5	eleq2d 2825	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}))
7		preq2 4666	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → {𝑋, 𝑛} = {𝑋, 𝑁})
8	7	sseq1d 3946	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ({𝑋, 𝑛} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ 𝑒))
9	8	rexbidv 3163	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
10	9	elrab 3629	. . . . . 6 ⊢ (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ (𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
11		eldifsn 4719	. . . . . . 7 ⊢ (𝑁 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋))
12	11	anbi1i 630	. . . . . 6 ⊢ ((𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
13	10, 12	bitri 276	. . . . 5 ⊢ (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
14	6, 13	bitrdi 288	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
15	14	pm5.32i 579	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ (𝑋 ∈ 𝑉 ∧ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
16		df-3an 1094	. . . 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 469	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋) ↔ (𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋)))
18		ancom 461	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ↔ (𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉))
19	18	anbi1i 630	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋))
20	17, 19	bitr3i 278	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋)) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋))
21	20	anbi1i 630	. . . 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 469	. . . 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 301	. . 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 276	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
25	3, 24	bitri 276	1 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∃wrex 3063  {crab 3391   ∖ cdif 3880   ⊆ wss 3883  {csn 4555  {cpr 4557  ‘cfv 6485  (class class class)co 7356  Vtxcvtx 29083  Edgcedg 29134   NeighbVtx cnbgr 29419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-nbgr 29420

This theorem is referenced by:  nbgrisvtx  29428  nbgr2vtx1edg  29437  nbuhgr2vtx1edgblem  29438  nbuhgr2vtx1edgb  29439  nbgrsym  29450  isuvtx  29482  iscplgredg  29504  cusgrexi  29530  structtocusgr  29533  dfvopnbgr2  48344