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Theorem dfnbgr3 26230
 Description: Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves (see also nbgrval 26228). (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 2621 . . . 4 (Edg‘𝐺) = (Edg‘𝐺)
31, 2nbgrval 26228 . . 3 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
43adantr 481 . 2 ((𝑁𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
5 edgval 25935 . . . . . 6 (Edg‘𝐺) = ran (iEdg‘𝐺)
6 dfnbgr3.i . . . . . . . 8 𝐼 = (iEdg‘𝐺)
76eqcomi 2630 . . . . . . 7 (iEdg‘𝐺) = 𝐼
87rneqi 5350 . . . . . 6 ran (iEdg‘𝐺) = ran 𝐼
95, 8eqtri 2643 . . . . 5 (Edg‘𝐺) = ran 𝐼
109rexeqi 3141 . . . 4 (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒)
11 funfn 5916 . . . . . . 7 (Fun 𝐼𝐼 Fn dom 𝐼)
1211biimpi 206 . . . . . 6 (Fun 𝐼𝐼 Fn dom 𝐼)
1312adantl 482 . . . . 5 ((𝑁𝑉 ∧ Fun 𝐼) → 𝐼 Fn dom 𝐼)
14 sseq2 3625 . . . . . 6 (𝑒 = (𝐼𝑖) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ (𝐼𝑖)))
1514rexrn 6359 . . . . 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 3187 . 2 ((𝑁𝑉 ∧ Fun 𝐼) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼𝑖)})
194, 18eqtrd 2655 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 384   = wceq 1482   ∈ wcel 1989  ∃wrex 2912  {crab 2915   ∖ cdif 3569   ⊆ wss 3572  {csn 4175  {cpr 4177  dom cdm 5112  ran crn 5113  Fun wfun 5880   Fn wfn 5881  ‘cfv 5886  (class class class)co 6647  Vtxcvtx 25868  iEdgciedg 25869  Edgcedg 25933   NeighbVtx cnbgr 26218 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-iota 5849  df-fun 5888  df-fn 5889  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-edg 25934  df-nbgr 26222 This theorem is referenced by: (None)
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