Theorem cusgrexi 27228

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 27226	. 2 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph)
3	1	cusgrexilem1 27224	. . . . . . . . 9 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃) ∈ V)
4		opvtxfv 26792	. . . . . . . . . 10 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
5	4	eqcomd 2830	. . . . . . . . 9 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑃) ∈ V) → 𝑉 = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
6	3, 5	mpdan 685	. . . . . . . 8 ⊢ (𝑉 ∈ 𝑊 → 𝑉 = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
7	6	eleq2d 2901	. . . . . . 7 ⊢ (𝑉 ∈ 𝑊 → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
8	7	biimpa 479	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
9		eldifi 4106	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛 ∈ 𝑉)
10	9	adantl 484	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ 𝑉)
11	3, 4	mpdan 685	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
12	11	eleq2d 2901	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝑊 → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑛 ∈ 𝑉))
13	12	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑛 ∈ 𝑉))
14	10, 13	mpbird 259	. . . . . . . . . 10 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
15		simplr 767	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑣 ∈ 𝑉)
16	11	eleq2d 2901	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝑊 → (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑣 ∈ 𝑉))
17	16	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑣 ∈ 𝑉))
18	15, 17	mpbird 259	. . . . . . . . . 10 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
19	14, 18	jca 514	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∧ 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
20		eldifsni 4725	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛 ≠ 𝑣)
21	20	adantl 484	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ≠ 𝑣)
22	1	cusgrexilem2 27227	. . . . . . . . . 10 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒)
23		edgval 26837	. . . . . . . . . . . . 13 ⊢ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩)
24		opiedgfv 26795	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
25	3, 24	mpdan 685	. . . . . . . . . . . . . 14 ⊢ (𝑉 ∈ 𝑊 → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
26	25	rneqd 5811	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ 𝑊 → ran (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran ( I ↾ 𝑃))
27	23, 26	syl5eq 2871	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝑊 → (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran ( I ↾ 𝑃))
28	27	rexeqdv 3419	. . . . . . . . . . 11 ⊢ (𝑉 ∈ 𝑊 → (∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒))
29	28	ad2antrr 724	. . . . . . . . . 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 259	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒)
31		eqid 2824	. . . . . . . . . 10 ⊢ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)
32		eqid 2824	. . . . . . . . . 10 ⊢ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩)
33	31, 32	nbgrel 27125	. . . . . . . . 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 1339	. . . . . . . 8 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
35	34	ralrimiva 3185	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
36	11	adantr 483	. . . . . . . . 9 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
37	36	difeq1d 4101	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣}) = (𝑉 ∖ {𝑣}))
38	37	raleqdv 3418	. . . . . . 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 259	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → ∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
40	31	uvtxel 27173	. . . . . 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 585	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
42	41	ralrimiva 3185	. . . 4 ⊢ (𝑉 ∈ 𝑊 → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
43	11	raleqdv 3418	. . . 4 ⊢ (𝑉 ∈ 𝑊 → (∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
44	42, 43	mpbird 259	. . 3 ⊢ (𝑉 ∈ 𝑊 → ∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
45		opex 5359	. . . 4 ⊢ ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ V
46	31	iscplgr 27200	. . . 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 259	. 2 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph)
49		iscusgr 27203	. 2 ⊢ (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph ↔ (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph ∧ ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph))
50	2, 48, 49	sylanbrc 585	1 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113   ≠ wne 3019  ∀wral 3141  ∃wrex 3142  {crab 3145  Vcvv 3497   ∖ cdif 3936   ⊆ wss 3939  𝒫 cpw 4542  {csn 4570  {cpr 4572  ⟨cop 4576   I cid 5462  ran crn 5559   ↾ cres 5560  ‘cfv 6358  (class class class)co 7159  2c2 11695  ♯chash 13693  Vtxcvtx 26784  iEdgciedg 26785  Edgcedg 26835  USGraphcusgr 26937   NeighbVtx cnbgr 27117  UnivVtxcuvtx 27170  ComplGraphccplgr 27194  ComplUSGraphccusgr 27195

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-dju 9333  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-2 11703  df-n0 11901  df-z 11985  df-uz 12247  df-fz 12896  df-hash 13694  df-vtx 26786  df-iedg 26787  df-edg 26836  df-usgr 26939  df-nbgr 27118  df-uvtx 27171  df-cplgr 27196  df-cusgr 27197

This theorem is referenced by:  cusgrexg  27229