Theorem nbgraop 25952

Description: The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens, 7-Oct-2017.)

Assertion

Ref	Expression
nbgraop	⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})

Distinct variable groups:   𝑛,𝑉   𝑛,𝐸   𝑛,𝑁   𝑛,𝑌   𝑛,𝑍

Proof of Theorem nbgraop

Dummy variables 𝑔 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nbgra 25949	. 2 ⊢ Neighbors = (𝑔 ∈ V, 𝑘 ∈ (1st ‘𝑔) ↦ {𝑛 ∈ (1st ‘𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd ‘𝑔)})
2		opex 4859	. . . 4 ⊢ ⟨𝑉, 𝐸⟩ ∈ V
3	2	a1i 11	. . 3 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ⟨𝑉, 𝐸⟩ ∈ V)
4		op1stg 7071	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) → (1st ‘⟨𝑉, 𝐸⟩) = 𝑉)
5	4	eqcomd 2616	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) → 𝑉 = (1st ‘⟨𝑉, 𝐸⟩))
6	5	eleq2d 2673	. . . . . 6 ⊢ ((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (1st ‘⟨𝑉, 𝐸⟩)))
7	6	biimpa 500	. . . . 5 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ (1st ‘⟨𝑉, 𝐸⟩))
8	7	adantr 480	. . . 4 ⊢ ((((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) ∧ 𝑔 = ⟨𝑉, 𝐸⟩) → 𝑁 ∈ (1st ‘⟨𝑉, 𝐸⟩))
9		fveq2 6103	. . . . 5 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → (1st ‘𝑔) = (1st ‘⟨𝑉, 𝐸⟩))
10	9	adantl 481	. . . 4 ⊢ ((((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) ∧ 𝑔 = ⟨𝑉, 𝐸⟩) → (1st ‘𝑔) = (1st ‘⟨𝑉, 𝐸⟩))
11	8, 10	eleqtrrd 2691	. . 3 ⊢ ((((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) ∧ 𝑔 = ⟨𝑉, 𝐸⟩) → 𝑁 ∈ (1st ‘𝑔))
12		fvex 6113	. . . 4 ⊢ (1st ‘𝑔) ∈ V
13		rabexg 4739	. . . 4 ⊢ ((1st ‘𝑔) ∈ V → {𝑛 ∈ (1st ‘𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd ‘𝑔)} ∈ V)
14	12, 13	mp1i 13	. . 3 ⊢ ((((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → {𝑛 ∈ (1st ‘𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd ‘𝑔)} ∈ V)
15	9, 4	sylan9eq 2664	. . . . . . . . 9 ⊢ ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ (𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍)) → (1st ‘𝑔) = 𝑉)
16	15	ex 449	. . . . . . . 8 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → ((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) → (1st ‘𝑔) = 𝑉))
17	16	adantr 480	. . . . . . 7 ⊢ ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → ((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) → (1st ‘𝑔) = 𝑉))
18	17	com12 32	. . . . . 6 ⊢ ((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) → ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → (1st ‘𝑔) = 𝑉))
19	18	adantr 480	. . . . 5 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → (1st ‘𝑔) = 𝑉))
20	19	imp 444	. . . 4 ⊢ ((((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → (1st ‘𝑔) = 𝑉)
21		preq1 4212	. . . . . . 7 ⊢ (𝑘 = 𝑁 → {𝑘, 𝑛} = {𝑁, 𝑛})
22	21	adantl 481	. . . . . 6 ⊢ ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → {𝑘, 𝑛} = {𝑁, 𝑛})
23	22	adantl 481	. . . . 5 ⊢ ((((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → {𝑘, 𝑛} = {𝑁, 𝑛})
24		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → (2nd ‘𝑔) = (2nd ‘⟨𝑉, 𝐸⟩))
25		op2ndg 7072	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
26	24, 25	sylan9eq 2664	. . . . . . . . . . 11 ⊢ ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ (𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍)) → (2nd ‘𝑔) = 𝐸)
27	26	ex 449	. . . . . . . . . 10 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → ((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) → (2nd ‘𝑔) = 𝐸))
28	27	adantr 480	. . . . . . . . 9 ⊢ ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → ((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) → (2nd ‘𝑔) = 𝐸))
29	28	com12 32	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) → ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → (2nd ‘𝑔) = 𝐸))
30	29	adantr 480	. . . . . . 7 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → (2nd ‘𝑔) = 𝐸))
31	30	imp 444	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → (2nd ‘𝑔) = 𝐸)
32	31	rneqd 5274	. . . . 5 ⊢ ((((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → ran (2nd ‘𝑔) = ran 𝐸)
33	23, 32	eleq12d 2682	. . . 4 ⊢ ((((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → ({𝑘, 𝑛} ∈ ran (2nd ‘𝑔) ↔ {𝑁, 𝑛} ∈ ran 𝐸))
34	20, 33	rabeqbidv 3168	. . 3 ⊢ ((((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → {𝑛 ∈ (1st ‘𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd ‘𝑔)} = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
35	3, 11, 14, 34	ovmpt2dv2 6692	. 2 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ( Neighbors = (𝑔 ∈ V, 𝑘 ∈ (1st ‘𝑔) ↦ {𝑛 ∈ (1st ‘𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd ‘𝑔)}) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸}))
36	1, 35	mpi 20	1 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  {cpr 4127  ⟨cop 4131  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1^st c1st 7057  2^nd c2nd 7058   Neighbors cnbgra 25946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-nbgra 25949

This theorem is referenced by:  nbgraop1  25954  nbgrael  25955  nbusgra  25957  rusgraprop3  26470