Theorem nbgrel 29498

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 29493	. . 3 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ 𝑉)
3	2	pm4.71ri 568	. 2 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝑋)))
4		nbgrel.e	. . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺)
5	1, 4	nbgrval 29494	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝐺 NeighbVtx 𝑋) = {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒})
6	5	eleq2d 2847	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}))
7		preq2 4690	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → {𝑋, 𝑛} = {𝑋, 𝑁})
8	7	sseq1d 3965	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ({𝑋, 𝑛} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ 𝑒))
9	8	rexbidv 3185	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
10	9	elrab 3649	. . . . . 6 ⊢ (𝑁 ∈ {𝑛 ∈ (𝑉 ∖ {𝑋}) ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ (𝑁 ∈ (𝑉 ∖ {𝑋}) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
11		eldifsn 4743	. . . . . . 7 ⊢ (𝑁 ∈ (𝑉 ∖ {𝑋}) ↔ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋))
12	11	anbi1i 633	. . . . . 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 582	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝑋)) ↔ (𝑋 ∈ 𝑉 ∧ ((𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋) ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
16		df-3an 1099	. . . 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 472	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋) ↔ (𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝑋)))
18		ancom 464	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ↔ (𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉))
19	18	anbi1i 633	. . . . . 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 633	. . . 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 472	. . . 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 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085  {crab 3413   ∖ cdif 3899   ⊆ wss 3902  {csn 4579  {cpr 4581  ‘cfv 6516  (class class class)co 7391  Vtxcvtx 29154  Edgcedg 29205   NeighbVtx cnbgr 29490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-nbgr 29491

This theorem is referenced by:  nbgrisvtx  29499  nbgr2vtx1edg  29508  nbuhgr2vtx1edgblem  29509  nbuhgr2vtx1edgb  29510  nbgrsym  29521  isuvtx  29553  iscplgredg  29575  cusgrexi  29601  structtocusgr  29604  dfvopnbgr2  48436