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Theorem nbgraop 25690
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 25687 . 2 Neighbors = (𝑔 ∈ V, 𝑘 ∈ (1st𝑔) ↦ {𝑛 ∈ (1st𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd𝑔)})
2 opex 4757 . . . 4 𝑉, 𝐸⟩ ∈ V
32a1i 11 . . 3 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ⟨𝑉, 𝐸⟩ ∈ V)
4 op1stg 6946 . . . . . . . 8 ((𝑉𝑌𝐸𝑍) → (1st ‘⟨𝑉, 𝐸⟩) = 𝑉)
54eqcomd 2520 . . . . . . 7 ((𝑉𝑌𝐸𝑍) → 𝑉 = (1st ‘⟨𝑉, 𝐸⟩))
65eleq2d 2577 . . . . . 6 ((𝑉𝑌𝐸𝑍) → (𝑁𝑉𝑁 ∈ (1st ‘⟨𝑉, 𝐸⟩)))
76biimpa 499 . . . . 5 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑁 ∈ (1st ‘⟨𝑉, 𝐸⟩))
87adantr 479 . . . 4 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ 𝑔 = ⟨𝑉, 𝐸⟩) → 𝑁 ∈ (1st ‘⟨𝑉, 𝐸⟩))
9 fveq2 5987 . . . . 5 (𝑔 = ⟨𝑉, 𝐸⟩ → (1st𝑔) = (1st ‘⟨𝑉, 𝐸⟩))
109adantl 480 . . . 4 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ 𝑔 = ⟨𝑉, 𝐸⟩) → (1st𝑔) = (1st ‘⟨𝑉, 𝐸⟩))
118, 10eleqtrrd 2595 . . 3 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ 𝑔 = ⟨𝑉, 𝐸⟩) → 𝑁 ∈ (1st𝑔))
12 fvex 5997 . . . 4 (1st𝑔) ∈ V
13 rabexg 4638 . . . 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 2568 . . . . . . . . 9 ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ (𝑉𝑌𝐸𝑍)) → (1st𝑔) = 𝑉)
1615ex 448 . . . . . . . 8 (𝑔 = ⟨𝑉, 𝐸⟩ → ((𝑉𝑌𝐸𝑍) → (1st𝑔) = 𝑉))
1716adantr 479 . . . . . . 7 ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → ((𝑉𝑌𝐸𝑍) → (1st𝑔) = 𝑉))
1817com12 32 . . . . . 6 ((𝑉𝑌𝐸𝑍) → ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → (1st𝑔) = 𝑉))
1918adantr 479 . . . . 5 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → (1st𝑔) = 𝑉))
2019imp 443 . . . 4 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → (1st𝑔) = 𝑉)
21 preq1 4115 . . . . . . 7 (𝑘 = 𝑁 → {𝑘, 𝑛} = {𝑁, 𝑛})
2221adantl 480 . . . . . 6 ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → {𝑘, 𝑛} = {𝑁, 𝑛})
2322adantl 480 . . . . 5 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → {𝑘, 𝑛} = {𝑁, 𝑛})
24 fveq2 5987 . . . . . . . . . . . 12 (𝑔 = ⟨𝑉, 𝐸⟩ → (2nd𝑔) = (2nd ‘⟨𝑉, 𝐸⟩))
25 op2ndg 6947 . . . . . . . . . . . 12 ((𝑉𝑌𝐸𝑍) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
2624, 25sylan9eq 2568 . . . . . . . . . . 11 ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ (𝑉𝑌𝐸𝑍)) → (2nd𝑔) = 𝐸)
2726ex 448 . . . . . . . . . 10 (𝑔 = ⟨𝑉, 𝐸⟩ → ((𝑉𝑌𝐸𝑍) → (2nd𝑔) = 𝐸))
2827adantr 479 . . . . . . . . 9 ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → ((𝑉𝑌𝐸𝑍) → (2nd𝑔) = 𝐸))
2928com12 32 . . . . . . . 8 ((𝑉𝑌𝐸𝑍) → ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → (2nd𝑔) = 𝐸))
3029adantr 479 . . . . . . 7 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ((𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁) → (2nd𝑔) = 𝐸))
3130imp 443 . . . . . 6 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → (2nd𝑔) = 𝐸)
3231rneqd 5165 . . . . 5 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → ran (2nd𝑔) = ran 𝐸)
3323, 32eleq12d 2586 . . . 4 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → ({𝑘, 𝑛} ∈ ran (2nd𝑔) ↔ {𝑁, 𝑛} ∈ ran 𝐸))
3420, 33rabeqbidv 3072 . . 3 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ (𝑔 = ⟨𝑉, 𝐸⟩ ∧ 𝑘 = 𝑁)) → {𝑛 ∈ (1st𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd𝑔)} = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
353, 11, 14, 34ovmpt2dv2 6569 . 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 382   = wceq 1474  wcel 1938  {crab 2804  Vcvv 3077  {cpr 4030  cop 4034  ran crn 4933  cfv 5689  (class class class)co 6426  cmpt2 6428  1st c1st 6932  2nd c2nd 6933   Neighbors cnbgra 25684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-iota 5653  df-fun 5691  df-fv 5697  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-1st 6934  df-2nd 6935  df-nbgra 25687
This theorem is referenced by:  nbgraop1  25692  nbgrael  25693  nbusgra  25695  rusgraprop3  26208
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