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Theorem cusgrexi 26563
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 26561 . 2 (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph)
31cusgrexilem1 26559 . . . . . . . . 9 (𝑉𝑊 → ( I ↾ 𝑃) ∈ V)
4 opvtxfv 26094 . . . . . . . . . 10 ((𝑉𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
54eqcomd 2811 . . . . . . . . 9 ((𝑉𝑊 ∧ ( I ↾ 𝑃) ∈ V) → 𝑉 = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
63, 5mpdan 670 . . . . . . . 8 (𝑉𝑊𝑉 = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
76eleq2d 2870 . . . . . . 7 (𝑉𝑊 → (𝑣𝑉𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
87biimpa 464 . . . . . 6 ((𝑉𝑊𝑣𝑉) → 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
9 eldifi 3928 . . . . . . . . . . . 12 (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛𝑉)
109adantl 469 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛𝑉)
113, 4mpdan 670 . . . . . . . . . . . . 13 (𝑉𝑊 → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
1211eleq2d 2870 . . . . . . . . . . . 12 (𝑉𝑊 → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑛𝑉))
1312ad2antrr 708 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑛𝑉))
1410, 13mpbird 248 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
15 simplr 776 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑣𝑉)
1611eleq2d 2870 . . . . . . . . . . . 12 (𝑉𝑊 → (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑣𝑉))
1716ad2antrr 708 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑣𝑉))
1815, 17mpbird 248 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
1914, 18jca 503 . . . . . . . . 9 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∧ 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
20 eldifsni 4509 . . . . . . . . . 10 (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛𝑣)
2120adantl 469 . . . . . . . . 9 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛𝑣)
221cusgrexilem2 26562 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒)
23 edgval 26151 . . . . . . . . . . . . 13 (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩)
24 opiedgfv 26097 . . . . . . . . . . . . . . 15 ((𝑉𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
253, 24mpdan 670 . . . . . . . . . . . . . 14 (𝑉𝑊 → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
2625rneqd 5551 . . . . . . . . . . . . 13 (𝑉𝑊 → ran (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran ( I ↾ 𝑃))
2723, 26syl5eq 2851 . . . . . . . . . . . 12 (𝑉𝑊 → (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran ( I ↾ 𝑃))
2827rexeqdv 3333 . . . . . . . . . . 11 (𝑉𝑊 → (∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒))
2928ad2antrr 708 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒))
3022, 29mpbird 248 . . . . . . . . 9 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒)
31 eqid 2805 . . . . . . . . . 10 (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)
32 eqid 2805 . . . . . . . . . 10 (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩)
3331, 32nbgrel 26445 . . . . . . . . 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 1436 . . . . . . . 8 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
3534ralrimiva 3153 . . . . . . 7 ((𝑉𝑊𝑣𝑉) → ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
3611adantr 468 . . . . . . . . 9 ((𝑉𝑊𝑣𝑉) → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
3736difeq1d 3923 . . . . . . . 8 ((𝑉𝑊𝑣𝑉) → ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣}) = (𝑉 ∖ {𝑣}))
3837raleqdv 3332 . . . . . . 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 248 . . . . . 6 ((𝑉𝑊𝑣𝑉) → ∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
4031uvtxel 26503 . . . . . 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 574 . . . . 5 ((𝑉𝑊𝑣𝑉) → 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
4241ralrimiva 3153 . . . 4 (𝑉𝑊 → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
4311raleqdv 3332 . . . 4 (𝑉𝑊 → (∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
4442, 43mpbird 248 . . 3 (𝑉𝑊 → ∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
45 opex 5119 . . . 4 𝑉, ( I ↾ 𝑃)⟩ ∈ V
4631iscplgr 26534 . . . 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 248 . 2 (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph)
49 iscusgr 26538 . 2 (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph ↔ (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph ∧ ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph))
502, 48, 49sylanbrc 574 1 (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2158  wne 2977  wral 3095  wrex 3096  {crab 3099  Vcvv 3390  cdif 3763  wss 3766  𝒫 cpw 4348  {csn 4367  {cpr 4369  cop 4373   I cid 5215  ran crn 5309  cres 5310  cfv 6098  (class class class)co 6871  2c2 11352  chash 13333  Vtxcvtx 26084  iEdgciedg 26085  Edgcedg 26149  USGraphcusgr 26255   NeighbVtx cnbgr 26436  UnivVtxcuvtx 26499  ComplGraphccplgr 26528  ComplUSGraphccusgr 26529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-rep 4960  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176  ax-cnex 10274  ax-resscn 10275  ax-1cn 10276  ax-icn 10277  ax-addcl 10278  ax-addrcl 10279  ax-mulcl 10280  ax-mulrcl 10281  ax-mulcom 10282  ax-addass 10283  ax-mulass 10284  ax-distr 10285  ax-i2m1 10286  ax-1ne0 10287  ax-1rid 10288  ax-rnegex 10289  ax-rrecex 10290  ax-cnre 10291  ax-pre-lttri 10292  ax-pre-lttrn 10293  ax-pre-ltadd 10294  ax-pre-mulgt0 10295
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-nel 3081  df-ral 3100  df-rex 3101  df-reu 3102  df-rmo 3103  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4627  df-int 4666  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-tr 4943  df-id 5216  df-eprel 5221  df-po 5229  df-so 5230  df-fr 5267  df-we 5269  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-pred 5890  df-ord 5936  df-on 5937  df-lim 5938  df-suc 5939  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-riota 6832  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-om 7293  df-1st 7395  df-2nd 7396  df-wrecs 7639  df-recs 7701  df-rdg 7739  df-1o 7793  df-oadd 7797  df-er 7976  df-en 8190  df-dom 8191  df-sdom 8192  df-fin 8193  df-card 9045  df-cda 9272  df-pnf 10358  df-mnf 10359  df-xr 10360  df-ltxr 10361  df-le 10362  df-sub 10550  df-neg 10551  df-nn 11303  df-2 11360  df-n0 11556  df-z 11640  df-uz 11901  df-fz 12546  df-hash 13334  df-vtx 26086  df-iedg 26087  df-edg 26150  df-usgr 26257  df-nbgr 26437  df-uvtx 26500  df-cplgr 26530  df-cusgr 26531
This theorem is referenced by:  cusgrexg  26564
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