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Theorem cusgrexi 27140
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 (𝑉𝑊 → ⟨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.
StepHypRef Expression
1 usgrexi.p . . 3 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}
21usgrexi 27138 . 2 (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph)
31cusgrexilem1 27136 . . . . . . . . 9 (𝑉𝑊 → ( I ↾ 𝑃) ∈ V)
4 opvtxfv 26704 . . . . . . . . . 10 ((𝑉𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
54eqcomd 2830 . . . . . . . . 9 ((𝑉𝑊 ∧ ( I ↾ 𝑃) ∈ V) → 𝑉 = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
63, 5mpdan 683 . . . . . . . 8 (𝑉𝑊𝑉 = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
76eleq2d 2902 . . . . . . 7 (𝑉𝑊 → (𝑣𝑉𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
87biimpa 477 . . . . . 6 ((𝑉𝑊𝑣𝑉) → 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
9 eldifi 4106 . . . . . . . . . . . 12 (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛𝑉)
109adantl 482 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛𝑉)
113, 4mpdan 683 . . . . . . . . . . . . 13 (𝑉𝑊 → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
1211eleq2d 2902 . . . . . . . . . . . 12 (𝑉𝑊 → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑛𝑉))
1312ad2antrr 722 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑛𝑉))
1410, 13mpbird 258 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
15 simplr 765 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑣𝑉)
1611eleq2d 2902 . . . . . . . . . . . 12 (𝑉𝑊 → (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑣𝑉))
1716ad2antrr 722 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑣𝑉))
1815, 17mpbird 258 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
1914, 18jca 512 . . . . . . . . 9 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∧ 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
20 eldifsni 4720 . . . . . . . . . 10 (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛𝑣)
2120adantl 482 . . . . . . . . 9 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛𝑣)
221cusgrexilem2 27139 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒)
23 edgval 26749 . . . . . . . . . . . . 13 (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩)
24 opiedgfv 26707 . . . . . . . . . . . . . . 15 ((𝑉𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
253, 24mpdan 683 . . . . . . . . . . . . . 14 (𝑉𝑊 → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
2625rneqd 5806 . . . . . . . . . . . . 13 (𝑉𝑊 → ran (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran ( I ↾ 𝑃))
2723, 26syl5eq 2872 . . . . . . . . . . . 12 (𝑉𝑊 → (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran ( I ↾ 𝑃))
2827rexeqdv 3421 . . . . . . . . . . 11 (𝑉𝑊 → (∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒))
2928ad2antrr 722 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒))
3022, 29mpbird 258 . . . . . . . . 9 (((𝑉𝑊𝑣𝑉) ∧ 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)⟩)
3331, 32nbgrel 27037 . . . . . . . . 9 (𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣) ↔ ((𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∧ 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)) ∧ 𝑛𝑣 ∧ ∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒))
3419, 21, 30, 33syl3anbrc 1337 . . . . . . . 8 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
3534ralrimiva 3186 . . . . . . 7 ((𝑉𝑊𝑣𝑉) → ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
3611adantr 481 . . . . . . . . 9 ((𝑉𝑊𝑣𝑉) → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
3736difeq1d 4101 . . . . . . . 8 ((𝑉𝑊𝑣𝑉) → ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣}) = (𝑉 ∖ {𝑣}))
3837raleqdv 3420 . . . . . . 7 ((𝑉𝑊𝑣𝑉) → (∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣)))
3935, 38mpbird 258 . . . . . 6 ((𝑉𝑊𝑣𝑉) → ∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
4031uvtxel 27085 . . . . . 6 (𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∧ ∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣)))
418, 39, 40sylanbrc 583 . . . . 5 ((𝑉𝑊𝑣𝑉) → 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
4241ralrimiva 3186 . . . 4 (𝑉𝑊 → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
4311raleqdv 3420 . . . 4 (𝑉𝑊 → (∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
4442, 43mpbird 258 . . 3 (𝑉𝑊 → ∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
45 opex 5352 . . . 4 𝑉, ( I ↾ 𝑃)⟩ ∈ V
4631iscplgr 27112 . . . 4 (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ V → (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
4745, 46mp1i 13 . . 3 (𝑉𝑊 → (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
4844, 47mpbird 258 . 2 (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph)
49 iscusgr 27115 . 2 (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph ↔ (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph ∧ ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph))
502, 48, 49sylanbrc 583 1 (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2106  wne 3020  wral 3142  wrex 3143  {crab 3146  Vcvv 3499  cdif 3936  wss 3939  𝒫 cpw 4541  {csn 4563  {cpr 4565  cop 4569   I cid 5457  ran crn 5554  cres 5555  cfv 6351  (class class class)co 7151  2c2 11684  chash 13683  Vtxcvtx 26696  iEdgciedg 26697  Edgcedg 26747  USGraphcusgr 26849   NeighbVtx cnbgr 27029  UnivVtxcuvtx 27082  ComplGraphccplgr 27106  ComplUSGraphccusgr 27107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  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 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12886  df-hash 13684  df-vtx 26698  df-iedg 26699  df-edg 26748  df-usgr 26851  df-nbgr 27030  df-uvtx 27083  df-cplgr 27108  df-cusgr 27109
This theorem is referenced by:  cusgrexg  27141
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