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Theorem dfnbgr3 26430
Description: Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves (see also nbgrval 26428). (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 2760 . . . 4 (Edg‘𝐺) = (Edg‘𝐺)
31, 2nbgrval 26428 . . 3 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
43adantr 472 . 2 ((𝑁𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
5 edgval 26140 . . . . . 6 (Edg‘𝐺) = ran (iEdg‘𝐺)
6 dfnbgr3.i . . . . . . . 8 𝐼 = (iEdg‘𝐺)
76eqcomi 2769 . . . . . . 7 (iEdg‘𝐺) = 𝐼
87rneqi 5507 . . . . . 6 ran (iEdg‘𝐺) = ran 𝐼
95, 8eqtri 2782 . . . . 5 (Edg‘𝐺) = ran 𝐼
109rexeqi 3282 . . . 4 (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒)
11 funfn 6079 . . . . . . 7 (Fun 𝐼𝐼 Fn dom 𝐼)
1211biimpi 206 . . . . . 6 (Fun 𝐼𝐼 Fn dom 𝐼)
1312adantl 473 . . . . 5 ((𝑁𝑉 ∧ Fun 𝐼) → 𝐼 Fn dom 𝐼)
14 sseq2 3768 . . . . . 6 (𝑒 = (𝐼𝑖) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ (𝐼𝑖)))
1514rexrn 6524 . . . . 5 (𝐼 Fn dom 𝐼 → (∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)))
1613, 15syl 17 . . . 4 ((𝑁𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)))
1710, 16syl5bb 272 . . 3 ((𝑁𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)))
1817rabbidv 3329 . 2 ((𝑁𝑉 ∧ Fun 𝐼) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)})
194, 18eqtrd 2794 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 196  wa 383   = wceq 1632  wcel 2139  wrex 3051  {crab 3054  cdif 3712  wss 3715  {csn 4321  {cpr 4323  dom cdm 5266  ran crn 5267  Fun wfun 6043   Fn wfn 6044  cfv 6049  (class class class)co 6813  Vtxcvtx 26073  iEdgciedg 26074  Edgcedg 26138   NeighbVtx cnbgr 26423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-edg 26139  df-nbgr 26424
This theorem is referenced by: (None)
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