Theorem cusgrexi 27810

Description: An arbitrary set 𝑉 regarded as set of vertices together with the set of pairs of elements of this set regarded as edges is a complete simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.) (Proof shortened by AV, 14-Feb-2022.)

Hypothesis

Ref	Expression
usgrexi.p	⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}

Assertion

Ref	Expression
cusgrexi	⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph)

Distinct variable groups:   𝑥,𝑉   𝑥,𝑃   𝑥,𝑊

Proof of Theorem cusgrexi

Dummy variables 𝑒 𝑛 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrexi.p	. . 3 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
2	1	usgrexi 27808	. 2 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph)
3	1	cusgrexilem1 27806	. . . . . . . . 9 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃) ∈ V)
4		opvtxfv 27374	. . . . . . . . . 10 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
5	4	eqcomd 2744	. . . . . . . . 9 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑃) ∈ V) → 𝑉 = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
6	3, 5	mpdan 684	. . . . . . . 8 ⊢ (𝑉 ∈ 𝑊 → 𝑉 = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
7	6	eleq2d 2824	. . . . . . 7 ⊢ (𝑉 ∈ 𝑊 → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
8	7	biimpa 477	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
9		eldifi 4061	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛 ∈ 𝑉)
10	9	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ 𝑉)
11	3, 4	mpdan 684	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
12	11	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝑊 → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑛 ∈ 𝑉))
13	12	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑛 ∈ 𝑉))
14	10, 13	mpbird 256	. . . . . . . . . 10 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
15		simplr 766	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑣 ∈ 𝑉)
16	11	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝑊 → (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑣 ∈ 𝑉))
17	16	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑣 ∈ 𝑉))
18	15, 17	mpbird 256	. . . . . . . . . 10 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
19	14, 18	jca 512	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∧ 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
20		eldifsni 4723	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛 ≠ 𝑣)
21	20	adantl 482	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ≠ 𝑣)
22	1	cusgrexilem2 27809	. . . . . . . . . 10 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒)
23		edgval 27419	. . . . . . . . . . . . 13 ⊢ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩)
24		opiedgfv 27377	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
25	3, 24	mpdan 684	. . . . . . . . . . . . . 14 ⊢ (𝑉 ∈ 𝑊 → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
26	25	rneqd 5847	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ 𝑊 → ran (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran ( I ↾ 𝑃))
27	23, 26	eqtrid 2790	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝑊 → (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran ( I ↾ 𝑃))
28	27	rexeqdv 3349	. . . . . . . . . . 11 ⊢ (𝑉 ∈ 𝑊 → (∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒))
29	28	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒))
30	22, 29	mpbird 256	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒)
31		eqid 2738	. . . . . . . . . 10 ⊢ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)
32		eqid 2738	. . . . . . . . . 10 ⊢ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩)
33	31, 32	nbgrel 27707	. . . . . . . . 9 ⊢ (𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣) ↔ ((𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∧ 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)) ∧ 𝑛 ≠ 𝑣 ∧ ∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒))
34	19, 21, 30, 33	syl3anbrc 1342	. . . . . . . 8 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
35	34	ralrimiva 3103	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
36	11	adantr 481	. . . . . . . . 9 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
37	36	difeq1d 4056	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣}) = (𝑉 ∖ {𝑣}))
38	37	raleqdv 3348	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → (∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣)))
39	35, 38	mpbird 256	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → ∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
40	31	uvtxel 27755	. . . . . 6 ⊢ (𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∧ ∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣)))
41	8, 39, 40	sylanbrc 583	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
42	41	ralrimiva 3103	. . . 4 ⊢ (𝑉 ∈ 𝑊 → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
43	11	raleqdv 3348	. . . 4 ⊢ (𝑉 ∈ 𝑊 → (∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
44	42, 43	mpbird 256	. . 3 ⊢ (𝑉 ∈ 𝑊 → ∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
45		opex 5379	. . . 4 ⊢ ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ V
46	31	iscplgr 27782	. . . 4 ⊢ (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ V → (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
47	45, 46	mp1i 13	. . 3 ⊢ (𝑉 ∈ 𝑊 → (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
48	44, 47	mpbird 256	. 2 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph)
49		iscusgr 27785	. 2 ⊢ (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph ↔ (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph ∧ ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph))
50	2, 48, 49	sylanbrc 583	1 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  𝒫 cpw 4533  {csn 4561  {cpr 4563  ⟨cop 4567   I cid 5488  ran crn 5590   ↾ cres 5591  ‘cfv 6433  (class class class)co 7275  2c2 12028  ♯chash 14044  Vtxcvtx 27366  iEdgciedg 27367  Edgcedg 27417  USGraphcusgr 27519   NeighbVtx cnbgr 27699  UnivVtxcuvtx 27752  ComplGraphccplgr 27776  ComplUSGraphccusgr 27777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-hash 14045  df-vtx 27368  df-iedg 27369  df-edg 27418  df-usgr 27521  df-nbgr 27700  df-uvtx 27753  df-cplgr 27778  df-cusgr 27779

This theorem is referenced by:  cusgrexg  27811