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Theorem dfnbgr3 40562
 Description: Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves [see also nbgraop1 25954]. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 25-Oct-2020.) (Revised by AV, 21-Mar-2021.)
Hypotheses
Ref Expression
dfnbgr3.v 𝑉 = (Vtx‘𝐺)
dfnbgr3.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
dfnbgr3 ((𝑁𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)})
Distinct variable groups:   𝑛,𝐺   𝑖,𝐼,𝑛   𝑖,𝑁,𝑛   𝑛,𝑉
Allowed substitution hints:   𝐺(𝑖)   𝑉(𝑖)

Proof of Theorem dfnbgr3
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 dfnbgr3.v . . . 4 𝑉 = (Vtx‘𝐺)
2 eqid 2610 . . . 4 (Edg‘𝐺) = (Edg‘𝐺)
31, 2nbgrval 40560 . . 3 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
43adantr 480 . 2 ((𝑁𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
511vgrex 25679 . . . . . 6 (𝑁𝑉𝐺 ∈ V)
65adantr 480 . . . . 5 ((𝑁𝑉 ∧ Fun 𝐼) → 𝐺 ∈ V)
7 edgaval 25794 . . . . . . 7 (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
8 dfnbgr3.i . . . . . . . . 9 𝐼 = (iEdg‘𝐺)
98eqcomi 2619 . . . . . . . 8 (iEdg‘𝐺) = 𝐼
109rneqi 5273 . . . . . . 7 ran (iEdg‘𝐺) = ran 𝐼
117, 10syl6eq 2660 . . . . . 6 (𝐺 ∈ V → (Edg‘𝐺) = ran 𝐼)
1211rexeqdv 3122 . . . . 5 (𝐺 ∈ V → (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒))
136, 12syl 17 . . . 4 ((𝑁𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒))
14 funfn 5833 . . . . . . 7 (Fun 𝐼𝐼 Fn dom 𝐼)
1514biimpi 205 . . . . . 6 (Fun 𝐼𝐼 Fn dom 𝐼)
1615adantl 481 . . . . 5 ((𝑁𝑉 ∧ Fun 𝐼) → 𝐼 Fn dom 𝐼)
17 sseq2 3590 . . . . . 6 (𝑒 = (𝐼𝑖) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ (𝐼𝑖)))
1817rexrn 6269 . . . . 5 (𝐼 Fn dom 𝐼 → (∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)))
1916, 18syl 17 . . . 4 ((𝑁𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)))
2013, 19bitrd 267 . . 3 ((𝑁𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)))
2120rabbidv 3164 . 2 ((𝑁𝑉 ∧ Fun 𝐼) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)})
224, 21eqtrd 2644 1 ((𝑁𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673  iEdgciedg 25674  Edgcedga 25792   NeighbVtx cnbgr 40550 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-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-edga 25793  df-nbgr 40554 This theorem is referenced by: (None)
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