Theorem nbgrel 29476

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 29471	. . 3 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ 𝑉)
3	2	pm4.71ri 567	. 2 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝑋)))
4		nbgrel.e	. . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺)
5	1, 4	nbgrval 29472	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝐺 NeighbVtx 𝑋) = {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒})
6	5	eleq2d 2838	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}))
7		preq2 4683	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → {𝑋, 𝑛} = {𝑋, 𝑁})
8	7	sseq1d 3958	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ({𝑋, 𝑛} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ 𝑒))
9	8	rexbidv 3176	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
10	9	elrab 3641	. . . . . 6 ⊢ (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ (𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
11		eldifsn 4736	. . . . . . 7 ⊢ (𝑁 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋))
12	11	anbi1i 632	. . . . . 6 ⊢ ((𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
13	10, 12	bitri 277	. . . . 5 ⊢ (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
14	6, 13	bitrdi 289	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
15	14	pm5.32i 581	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ (𝑋 ∈ 𝑉 ∧ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
16		df-3an 1097	. . . 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 471	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋) ↔ (𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋)))
18		ancom 463	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ↔ (𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉))
19	18	anbi1i 632	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋))
20	17, 19	bitr3i 279	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋)) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋))
21	20	anbi1i 632	. . . 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 471	. . . 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 302	. . 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 277	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
25	3, 24	bitri 277	1 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  ∃wrex 3076  {crab 3404   ∖ cdif 3892   ⊆ wss 3895  {csn 4572  {cpr 4574  ‘cfv 6506  (class class class)co 7381  Vtxcvtx 29132  Edgcedg 29183   NeighbVtx cnbgr 29468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-1st 7955  df-2nd 7956  df-nbgr 29469

This theorem is referenced by:  nbgrisvtx  29477  nbgr2vtx1edg  29486  nbuhgr2vtx1edgblem  29487  nbuhgr2vtx1edgb  29488  nbgrsym  29499  isuvtx  29531  iscplgredg  29553  cusgrexi  29579  structtocusgr  29582  dfvopnbgr2  48413