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Theorem dfnbgr3 26635
Description: Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves (see also nbgrval 26633). (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 2825 . . . 4 (Edg‘𝐺) = (Edg‘𝐺)
31, 2nbgrval 26633 . . 3 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
43adantr 474 . 2 ((𝑁𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
5 edgval 26347 . . . . . 6 (Edg‘𝐺) = ran (iEdg‘𝐺)
6 dfnbgr3.i . . . . . . . 8 𝐼 = (iEdg‘𝐺)
76eqcomi 2834 . . . . . . 7 (iEdg‘𝐺) = 𝐼
87rneqi 5584 . . . . . 6 ran (iEdg‘𝐺) = ran 𝐼
95, 8eqtri 2849 . . . . 5 (Edg‘𝐺) = ran 𝐼
109rexeqi 3355 . . . 4 (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒)
11 funfn 6153 . . . . . . 7 (Fun 𝐼𝐼 Fn dom 𝐼)
1211biimpi 208 . . . . . 6 (Fun 𝐼𝐼 Fn dom 𝐼)
1312adantl 475 . . . . 5 ((𝑁𝑉 ∧ Fun 𝐼) → 𝐼 Fn dom 𝐼)
14 sseq2 3852 . . . . . 6 (𝑒 = (𝐼𝑖) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ (𝐼𝑖)))
1514rexrn 6610 . . . . 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 275 . . 3 ((𝑁𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)))
1817rabbidv 3402 . 2 ((𝑁𝑉 ∧ Fun 𝐼) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)})
194, 18eqtrd 2861 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 198  wa 386   = wceq 1658  wcel 2166  wrex 3118  {crab 3121  cdif 3795  wss 3798  {csn 4397  {cpr 4399  dom cdm 5342  ran crn 5343  Fun wfun 6117   Fn wfn 6118  cfv 6123  (class class class)co 6905  Vtxcvtx 26294  iEdgciedg 26295  Edgcedg 26345   NeighbVtx cnbgr 26629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fn 6126  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-edg 26346  df-nbgr 26630
This theorem is referenced by: (None)
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