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Theorem cusgrexi 26220
Description: An arbitrary set regarded as 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, 10-Nov-2021.)
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 26218 . 2 (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph )
31cusgrexilem1 26216 . . . . . . . . . 10 (𝑉𝑊 → ( I ↾ 𝑃) ∈ V)
4 opvtxfv 25779 . . . . . . . . . . 11 ((𝑉𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
54eqcomd 2632 . . . . . . . . . 10 ((𝑉𝑊 ∧ ( I ↾ 𝑃) ∈ V) → 𝑉 = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
63, 5mpdan 701 . . . . . . . . 9 (𝑉𝑊𝑉 = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
76eleq2d 2689 . . . . . . . 8 (𝑉𝑊 → (𝑣𝑉𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
87biimpa 501 . . . . . . 7 ((𝑉𝑊𝑣𝑉) → 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
9 eldifi 3715 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛𝑉)
109adantl 482 . . . . . . . . . . . 12 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛𝑉)
113, 4mpdan 701 . . . . . . . . . . . . . 14 (𝑉𝑊 → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
1211eleq2d 2689 . . . . . . . . . . . . 13 (𝑉𝑊 → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑛𝑉))
1312ad2antrr 761 . . . . . . . . . . . 12 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑛𝑉))
1410, 13mpbird 247 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
15 simplr 791 . . . . . . . . . . . 12 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑣𝑉)
1611eleq2d 2689 . . . . . . . . . . . . 13 (𝑉𝑊 → (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑣𝑉))
1716ad2antrr 761 . . . . . . . . . . . 12 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑣𝑉))
1815, 17mpbird 247 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
1914, 18jca 554 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∧ 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
20 eldifsni 4294 . . . . . . . . . . 11 (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛𝑣)
2120adantl 482 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛𝑣)
221cusgrexilem2 26219 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒)
23 opex 4898 . . . . . . . . . . . . . . 15 𝑉, ( I ↾ 𝑃)⟩ ∈ V
24 edgval 25836 . . . . . . . . . . . . . . 15 (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ V → (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩))
2523, 24mp1i 13 . . . . . . . . . . . . . 14 (𝑉𝑊 → (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩))
26 opiedgfv 25782 . . . . . . . . . . . . . . . 16 ((𝑉𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
273, 26mpdan 701 . . . . . . . . . . . . . . 15 (𝑉𝑊 → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
2827rneqd 5317 . . . . . . . . . . . . . 14 (𝑉𝑊 → ran (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran ( I ↾ 𝑃))
2925, 28eqtrd 2660 . . . . . . . . . . . . 13 (𝑉𝑊 → (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran ( I ↾ 𝑃))
3029rexeqdv 3139 . . . . . . . . . . . 12 (𝑉𝑊 → (∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒))
3130ad2antrr 761 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒))
3222, 31mpbird 247 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒)
33 eqid 2626 . . . . . . . . . . . 12 (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)
34 eqid 2626 . . . . . . . . . . . 12 (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩)
3533, 34nbgrel 26119 . . . . . . . . . . 11 (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ V → (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)))
3623, 35mp1i 13 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (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)))
3719, 21, 32, 36mpbir3and 1243 . . . . . . . . 9 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
3837ralrimiva 2965 . . . . . . . 8 ((𝑉𝑊𝑣𝑉) → ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
3911adantr 481 . . . . . . . . . 10 ((𝑉𝑊𝑣𝑉) → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
4039difeq1d 3710 . . . . . . . . 9 ((𝑉𝑊𝑣𝑉) → ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣}) = (𝑉 ∖ {𝑣}))
4140raleqdv 3138 . . . . . . . 8 ((𝑉𝑊𝑣𝑉) → (∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣)))
4238, 41mpbird 247 . . . . . . 7 ((𝑉𝑊𝑣𝑉) → ∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
438, 42jca 554 . . . . . 6 ((𝑉𝑊𝑣𝑉) → (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∧ ∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣)))
4433uvtxael 26169 . . . . . . 7 (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ V → (𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∧ ∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))))
4523, 44mp1i 13 . . . . . 6 ((𝑉𝑊𝑣𝑉) → (𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∧ ∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))))
4643, 45mpbird 247 . . . . 5 ((𝑉𝑊𝑣𝑉) → 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
4746ralrimiva 2965 . . . 4 (𝑉𝑊 → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
4811raleqdv 3138 . . . 4 (𝑉𝑊 → (∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
4947, 48mpbird 247 . . 3 (𝑉𝑊 → ∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
5033iscplgr 26191 . . . 4 (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ V → (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
5123, 50mp1i 13 . . 3 (𝑉𝑊 → (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
5249, 51mpbird 247 . 2 (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph)
53 iscusgr 26195 . 2 (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph ↔ (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph ∧ ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph))
542, 52, 53sylanbrc 697 1 (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wral 2912  wrex 2913  {crab 2916  Vcvv 3191  cdif 3557  wss 3560  𝒫 cpw 4135  {csn 4153  {cpr 4155  cop 4159   I cid 4989  ran crn 5080  cres 5081  cfv 5850  (class class class)co 6605  2c2 11015  #chash 13054  Vtxcvtx 25769  iEdgciedg 25770  Edgcedg 25834   USGraph cusgr 25932   NeighbVtx cnbgr 26105  UnivVtxcuvtxa 26106  ComplGraphccplgr 26107  ComplUSGraphccusgr 26108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-hash 13055  df-vtx 25771  df-iedg 25772  df-edg 25835  df-usgr 25934  df-nbgr 26109  df-uvtxa 26111  df-cplgr 26112  df-cusgr 26113
This theorem is referenced by:  cusgrexg  26221
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