Theorem dfnbgr3 29241

Description: Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves (see also nbgrval 29239). (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 2729	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	nbgrval 29239	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
4	3	adantr 480	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒})
5		edgval 28952	. . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
6		dfnbgr3.i	. . . . . . . 8 ⊢ 𝐼 = (iEdg‘𝐺)
7	6	eqcomi 2738	. . . . . . 7 ⊢ (iEdg‘𝐺) = 𝐼
8	7	rneqi 5890	. . . . . 6 ⊢ ran (iEdg‘𝐺) = ran 𝐼
9	5, 8	eqtri 2752	. . . . 5 ⊢ (Edg‘𝐺) = ran 𝐼
10	9	rexeqi 3295	. . . 4 ⊢ (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒)
11		funfn 6530	. . . . . . 7 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼)
12	11	biimpi 216	. . . . . 6 ⊢ (Fun 𝐼 → 𝐼 Fn dom 𝐼)
13	12	adantl 481	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → 𝐼 Fn dom 𝐼)
14		sseq2 3970	. . . . . 6 ⊢ (𝑒 = (𝐼‘𝑖) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
15	14	rexrn 7041	. . . . 5 ⊢ (𝐼 Fn dom 𝐼 → (∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
16	13, 15	syl 17	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ ran 𝐼{𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
17	10, 16	bitrid 283	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)))
18	17	rabbidv 3410	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)})
19	4, 18	eqtrd 2764	1 ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3402   ∖ cdif 3908   ⊆ wss 3911  {csn 4585  {cpr 4587  dom cdm 5631  ran crn 5632  Fun wfun 6493   Fn wfn 6494  ‘cfv 6499  (class class class)co 7369  Vtxcvtx 28899  iEdgciedg 28900  Edgcedg 28950   NeighbVtx cnbgr 29235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fn 6502  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-edg 28951  df-nbgr 29236

This theorem is referenced by: (None)