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	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸)))

Proof of Theorem nbgrael

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

Step	Hyp	Ref	Expression
1		df-nbgra 25949	. . . 4 ⊢ Neighbors = (𝑔 ∈ V, 𝑘 ∈ (1st ‘𝑔) ↦ {𝑛 ∈ (1st ‘𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd ‘𝑔)})
2	1	mpt2xopn0yelv 7226	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) → 𝐾 ∈ 𝑉))
3	2	pm4.71rd 665	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾))))
4		nbgraop 25952	. . . . . 6 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝐾) = {𝑛 ∈ 𝑉 ∣ {𝐾, 𝑛} ∈ ran 𝐸})
5	4	eleq2d 2673	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ 𝑁 ∈ {𝑛 ∈ 𝑉 ∣ {𝐾, 𝑛} ∈ ran 𝐸}))
6		preq2 4213	. . . . . . 7 ⊢ (𝑛 = 𝑁 → {𝐾, 𝑛} = {𝐾, 𝑁})
7	6	eleq1d 2672	. . . . . 6 ⊢ (𝑛 = 𝑁 → ({𝐾, 𝑛} ∈ ran 𝐸 ↔ {𝐾, 𝑁} ∈ ran 𝐸))
8	7	elrab 3331	. . . . 5 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ {𝐾, 𝑛} ∈ ran 𝐸} ↔ (𝑁 ∈ 𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸))
9	5, 8	syl6bb 275	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ (𝑁 ∈ 𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸)))
10	9	pm5.32da 671	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾)) ↔ (𝐾 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸))))
11		3anass 1035	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸) ↔ (𝐾 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸)))
12	10, 11	syl6bbr 277	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾)) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸)))
13	3, 12	bitrd 267	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ {𝐾, 𝑁} ∈ ran 𝐸)))

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  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-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