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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 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {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.
StepHypRef Expression
1 df-nbgra 25949 . 2 Neighbors = (𝑔 ∈ V, 𝑘 ∈ (1st𝑔) ↦ {𝑛 ∈ (1st𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd𝑔)})
2 opex 4859 . . . 4 𝑉, 𝐸⟩ ∈ V
32a1i 11 . . 3 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ⟨𝑉, 𝐸⟩ ∈ V)
4 op1stg 7071 . . . . . . . 8 ((𝑉𝑌𝐸𝑍) → (1st ‘⟨𝑉, 𝐸⟩) = 𝑉)
54eqcomd 2616 . . . . . . 7 ((𝑉𝑌𝐸𝑍) → 𝑉 = (1st ‘⟨𝑉, 𝐸⟩))
65eleq2d 2673 . . . . . 6 ((𝑉𝑌𝐸𝑍) → (𝑁𝑉𝑁 ∈ (1st ‘⟨𝑉, 𝐸⟩)))
76biimpa 500 . . . . 5 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑁 ∈ (1st ‘⟨𝑉, 𝐸⟩))
87adantr 480 . . . 4 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ 𝑔 = ⟨𝑉, 𝐸⟩) → 𝑁 ∈ (1st ‘⟨𝑉, 𝐸⟩))
9 fveq2 6103 . . . . 5 (𝑔 = ⟨𝑉, 𝐸⟩ → (1st𝑔) = (1st ‘⟨𝑉, 𝐸⟩))
109adantl 481 . . . 4 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ 𝑔 = ⟨𝑉, 𝐸⟩) → (1st𝑔) = (1st ‘⟨𝑉, 𝐸⟩))
118, 10eleqtrrd 2691 . . 3 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ 𝑔 = ⟨𝑉, 𝐸⟩) → 𝑁 ∈ (1st𝑔))
12 fvex 6113 . . . 4 (1st𝑔) ∈ V
13 rabexg 4739 . . . 4 ((1st𝑔) ∈ V → {𝑛 ∈ (1st𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd𝑔)} ∈ V)
1412, 13mp1i 13 . . 3 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → {𝑛 ∈ (1st𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd𝑔)} ∈ V)
159, 4sylan9eq 2664 . . . . . . . . 9 ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ (𝑉𝑌𝐸𝑍)) → (1st𝑔) = 𝑉)
1615ex 449 . . . . . . . 8 (𝑔 = ⟨𝑉, 𝐸⟩ → ((𝑉𝑌𝐸𝑍) → (1st𝑔) = 𝑉))
1716adantr 480 . . . . . . 7 ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → ((𝑉𝑌𝐸𝑍) → (1st𝑔) = 𝑉))
1817com12 32 . . . . . 6 ((𝑉𝑌𝐸𝑍) → ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → (1st𝑔) = 𝑉))
1918adantr 480 . . . . 5 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → (1st𝑔) = 𝑉))
2019imp 444 . . . 4 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → (1st𝑔) = 𝑉)
21 preq1 4212 . . . . . . 7 (𝑘 = 𝑁 → {𝑘, 𝑛} = {𝑁, 𝑛})
2221adantl 481 . . . . . 6 ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → {𝑘, 𝑛} = {𝑁, 𝑛})
2322adantl 481 . . . . 5 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → {𝑘, 𝑛} = {𝑁, 𝑛})
24 fveq2 6103 . . . . . . . . . . . 12 (𝑔 = ⟨𝑉, 𝐸⟩ → (2nd𝑔) = (2nd ‘⟨𝑉, 𝐸⟩))
25 op2ndg 7072 . . . . . . . . . . . 12 ((𝑉𝑌𝐸𝑍) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
2624, 25sylan9eq 2664 . . . . . . . . . . 11 ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ (𝑉𝑌𝐸𝑍)) → (2nd𝑔) = 𝐸)
2726ex 449 . . . . . . . . . 10 (𝑔 = ⟨𝑉, 𝐸⟩ → ((𝑉𝑌𝐸𝑍) → (2nd𝑔) = 𝐸))
2827adantr 480 . . . . . . . . 9 ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → ((𝑉𝑌𝐸𝑍) → (2nd𝑔) = 𝐸))
2928com12 32 . . . . . . . 8 ((𝑉𝑌𝐸𝑍) → ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → (2nd𝑔) = 𝐸))
3029adantr 480 . . . . . . 7 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → (2nd𝑔) = 𝐸))
3130imp 444 . . . . . 6 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → (2nd𝑔) = 𝐸)
3231rneqd 5274 . . . . 5 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → ran (2nd𝑔) = ran 𝐸)
3323, 32eleq12d 2682 . . . 4 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → ({𝑘, 𝑛} ∈ ran (2nd𝑔) ↔ {𝑁, 𝑛} ∈ ran 𝐸))
3420, 33rabeqbidv 3168 . . 3 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → {𝑛 ∈ (1st𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd𝑔)} = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
353, 11, 14, 34ovmpt2dv2 6692 . 2 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ( Neighbors = (𝑔 ∈ V, 𝑘 ∈ (1st𝑔) ↦ {𝑛 ∈ (1st𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd𝑔)}) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸}))
361, 35mpi 20 1 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
 Colors of variables: wff setvar class 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  1st c1st 7057  2nd 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
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