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Theorem nbgraop1 25954
 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, 17-Dec-2017.)
Assertion
Ref Expression
nbgraop1 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (Fun 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ ∃𝑖 ∈ dom 𝐸(𝐸𝑖) = {𝑁, 𝑛}}))
Distinct variable groups:   𝑖,𝐸,𝑛   𝑖,𝑁,𝑛   𝑖,𝑉,𝑛   𝑛,𝑌   𝑛,𝑍
Allowed substitution hints:   𝑌(𝑖)   𝑍(𝑖)

Proof of Theorem nbgraop1
StepHypRef Expression
1 nbgraop 25952 . . 3 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
2 elrnrexdmb 6272 . . . . 5 (Fun 𝐸 → ({𝑁, 𝑛} ∈ ran 𝐸 ↔ ∃𝑖 ∈ dom 𝐸{𝑁, 𝑛} = (𝐸𝑖)))
3 eqcom 2617 . . . . . 6 ({𝑁, 𝑛} = (𝐸𝑖) ↔ (𝐸𝑖) = {𝑁, 𝑛})
43rexbii 3023 . . . . 5 (∃𝑖 ∈ dom 𝐸{𝑁, 𝑛} = (𝐸𝑖) ↔ ∃𝑖 ∈ dom 𝐸(𝐸𝑖) = {𝑁, 𝑛})
52, 4syl6bb 275 . . . 4 (Fun 𝐸 → ({𝑁, 𝑛} ∈ ran 𝐸 ↔ ∃𝑖 ∈ dom 𝐸(𝐸𝑖) = {𝑁, 𝑛}))
65rabbidv 3164 . . 3 (Fun 𝐸 → {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} = {𝑛𝑉 ∣ ∃𝑖 ∈ dom 𝐸(𝐸𝑖) = {𝑁, 𝑛}})
71, 6sylan9eq 2664 . 2 ((((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) ∧ Fun 𝐸) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ ∃𝑖 ∈ dom 𝐸(𝐸𝑖) = {𝑁, 𝑛}})
87ex 449 1 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (Fun 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ ∃𝑖 ∈ dom 𝐸(𝐸𝑖) = {𝑁, 𝑛}}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  {cpr 4127  ⟨cop 4131  dom cdm 5038  ran crn 5039  Fun wfun 5798  ‘cfv 5804  (class class class)co 6549   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-fn 5807  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: (None)
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