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Theorem dfnbgr3 29269
Description: Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves (see also nbgrval 29267). (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 2726 . . . 4 (Edg‘𝐺) = (Edg‘𝐺)
31, 2nbgrval 29267 . . 3 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
43adantr 479 . 2 ((𝑁𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
5 edgval 28980 . . . . . 6 (Edg‘𝐺) = ran (iEdg‘𝐺)
6 dfnbgr3.i . . . . . . . 8 𝐼 = (iEdg‘𝐺)
76eqcomi 2735 . . . . . . 7 (iEdg‘𝐺) = 𝐼
87rneqi 5934 . . . . . 6 ran (iEdg‘𝐺) = ran 𝐼
95, 8eqtri 2754 . . . . 5 (Edg‘𝐺) = ran 𝐼
109rexeqi 3314 . . . 4 (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒)
11 funfn 6579 . . . . . . 7 (Fun 𝐼𝐼 Fn dom 𝐼)
1211biimpi 215 . . . . . 6 (Fun 𝐼𝐼 Fn dom 𝐼)
1312adantl 480 . . . . 5 ((𝑁𝑉 ∧ Fun 𝐼) → 𝐼 Fn dom 𝐼)
14 sseq2 4006 . . . . . 6 (𝑒 = (𝐼𝑖) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ (𝐼𝑖)))
1514rexrn 7091 . . . . 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, 16bitrid 282 . . 3 ((𝑁𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)))
1817rabbidv 3428 . 2 ((𝑁𝑉 ∧ Fun 𝐼) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)})
194, 18eqtrd 2766 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 205  wa 394   = wceq 1534  wcel 2099  wrex 3060  {crab 3420  cdif 3944  wss 3947  {csn 4624  {cpr 4626  dom cdm 5673  ran crn 5674  Fun wfun 6538   Fn wfn 6539  cfv 6544  (class class class)co 7414  Vtxcvtx 28927  iEdgciedg 28928  Edgcedg 28978   NeighbVtx cnbgr 29263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-ov 7417  df-oprab 7418  df-mpo 7419  df-edg 28979  df-nbgr 29264
This theorem is referenced by: (None)
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