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Theorem nbgrael 25955
 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 and Mario Carneiro, 9-Oct-2017.)
Assertion
Ref Expression
nbgrael ((𝑉𝑋𝐸𝑌) → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ (𝐾𝑉𝑁𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸)))

Proof of Theorem nbgrael
Dummy variables 𝑔 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nbgra 25949 . . . 4 Neighbors = (𝑔 ∈ V, 𝑘 ∈ (1st𝑔) ↦ {𝑛 ∈ (1st𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd𝑔)})
21mpt2xopn0yelv 7226 . . 3 ((𝑉𝑋𝐸𝑌) → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) → 𝐾𝑉))
32pm4.71rd 665 . 2 ((𝑉𝑋𝐸𝑌) → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ (𝐾𝑉𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾))))
4 nbgraop 25952 . . . . . 6 (((𝑉𝑋𝐸𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝐾) = {𝑛𝑉 ∣ {𝐾, 𝑛} ∈ ran 𝐸})
54eleq2d 2673 . . . . 5 (((𝑉𝑋𝐸𝑌) ∧ 𝐾𝑉) → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ 𝑁 ∈ {𝑛𝑉 ∣ {𝐾, 𝑛} ∈ ran 𝐸}))
6 preq2 4213 . . . . . . 7 (𝑛 = 𝑁 → {𝐾, 𝑛} = {𝐾, 𝑁})
76eleq1d 2672 . . . . . 6 (𝑛 = 𝑁 → ({𝐾, 𝑛} ∈ ran 𝐸 ↔ {𝐾, 𝑁} ∈ ran 𝐸))
87elrab 3331 . . . . 5 (𝑁 ∈ {𝑛𝑉 ∣ {𝐾, 𝑛} ∈ ran 𝐸} ↔ (𝑁𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸))
95, 8syl6bb 275 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ 𝐾𝑉) → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ (𝑁𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸)))
109pm5.32da 671 . . 3 ((𝑉𝑋𝐸𝑌) → ((𝐾𝑉𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾)) ↔ (𝐾𝑉 ∧ (𝑁𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸))))
11 3anass 1035 . . 3 ((𝐾𝑉𝑁𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸) ↔ (𝐾𝑉 ∧ (𝑁𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸)))
1210, 11syl6bbr 277 . 2 ((𝑉𝑋𝐸𝑌) → ((𝐾𝑉𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾)) ↔ (𝐾𝑉𝑁𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸)))
133, 12bitrd 267 1 ((𝑉𝑋𝐸𝑌) → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ (𝐾𝑉𝑁𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  {cpr 4127  ⟨cop 4131  ran crn 5039  ‘cfv 5804  (class class class)co 6549  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-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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-iun 4457  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-res 5050  df-ima 5051  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:  nbgrasym  25968  nbgraf1olem1  25970  usg2spot2nb  26592  extwwlkfablem1  26601
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