Theorem dfnbgr3 29628

Description: Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves (see also nbgrval 29626). (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	⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)})

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.

Step	Hyp	Ref	Expression
1		dfnbgr3.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		eqid 2769	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	nbgrval 29626	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
4	3	adantr 485	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
5		edgval 29339	. . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
6		dfnbgr3.i	. . . . . . . 8 ⊢ 𝐼 = (iEdg‘𝐺)
7	6	eqcomi 2778	. . . . . . 7 ⊢ (iEdg‘𝐺) = 𝐼
8	7	rneqi 5928	. . . . . 6 ⊢ ran (iEdg‘𝐺) = ran 𝐼
9	5, 8	eqtri 2792	. . . . 5 ⊢ (Edg‘𝐺) = ran 𝐼
10	9	rexeqi 3328	. . . 4 ⊢ (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒)
11		funfn 6567	. . . . . 6 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼)
12	11	bilani 509	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → 𝐼 Fn dom 𝐼)
13		sseq2 3971	. . . . . 6 ⊢ (𝑒 = (𝐼‘𝑖) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
14	13	rexrn 7083	. . . . 5 ⊢ (𝐼 Fn dom 𝐼 → (∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
15	12, 14	syl 18	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
16	10, 15	bitrid 286	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
17	16	rabbidv 3430	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)})
18	4, 17	eqtrd 2804	1 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  {crab 3423   ∖ cdif 3910   ⊆ wss 3913  {csn 4594  {cpr 4596  dom cdm 5662  ran crn 5663  Fun wfun 6531   Fn wfn 6532  ‘cfv 6537  (class class class)co 7411  Vtxcvtx 29286  iEdgciedg 29287  Edgcedg 29337   NeighbVtx cnbgr 29622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-edg 29338  df-nbgr 29623

This theorem is referenced by: (None)