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Theorem cusgrexi 26470
 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 26468 . 2 (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph)
31cusgrexilem1 26466 . . . . . . . . 9 (𝑉𝑊 → ( I ↾ 𝑃) ∈ V)
4 opvtxfv 26004 . . . . . . . . . 10 ((𝑉𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
54eqcomd 2730 . . . . . . . . 9 ((𝑉𝑊 ∧ ( I ↾ 𝑃) ∈ V) → 𝑉 = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
63, 5mpdan 705 . . . . . . . 8 (𝑉𝑊𝑉 = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
76eleq2d 2789 . . . . . . 7 (𝑉𝑊 → (𝑣𝑉𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
87biimpa 502 . . . . . 6 ((𝑉𝑊𝑣𝑉) → 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
9 eldifi 3840 . . . . . . . . . . . 12 (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛𝑉)
109adantl 473 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛𝑉)
113, 4mpdan 705 . . . . . . . . . . . . 13 (𝑉𝑊 → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
1211eleq2d 2789 . . . . . . . . . . . 12 (𝑉𝑊 → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑛𝑉))
1312ad2antrr 764 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑛𝑉))
1410, 13mpbird 247 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
15 simplr 809 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑣𝑉)
1611eleq2d 2789 . . . . . . . . . . . 12 (𝑉𝑊 → (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑣𝑉))
1716ad2antrr 764 . . . . . . . . . . 11 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑣𝑉))
1815, 17mpbird 247 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
1914, 18jca 555 . . . . . . . . 9 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∧ 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
20 eldifsni 4429 . . . . . . . . . 10 (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛𝑣)
2120adantl 473 . . . . . . . . 9 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛𝑣)
221cusgrexilem2 26469 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒)
23 edgval 26061 . . . . . . . . . . . . 13 (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩)
24 opiedgfv 26007 . . . . . . . . . . . . . . 15 ((𝑉𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
253, 24mpdan 705 . . . . . . . . . . . . . 14 (𝑉𝑊 → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
2625rneqd 5460 . . . . . . . . . . . . 13 (𝑉𝑊 → ran (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran ( I ↾ 𝑃))
2723, 26syl5eq 2770 . . . . . . . . . . . 12 (𝑉𝑊 → (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran ( I ↾ 𝑃))
2827rexeqdv 3248 . . . . . . . . . . 11 (𝑉𝑊 → (∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒))
2928ad2antrr 764 . . . . . . . . . 10 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒))
3022, 29mpbird 247 . . . . . . . . 9 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒)
31 eqid 2724 . . . . . . . . . 10 (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)
32 eqid 2724 . . . . . . . . . 10 (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩)
3331, 32nbgrel 26353 . . . . . . . . 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 1383 . . . . . . . 8 (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
3534ralrimiva 3068 . . . . . . 7 ((𝑉𝑊𝑣𝑉) → ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
3611adantr 472 . . . . . . . . 9 ((𝑉𝑊𝑣𝑉) → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
3736difeq1d 3835 . . . . . . . 8 ((𝑉𝑊𝑣𝑉) → ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣}) = (𝑉 ∖ {𝑣}))
3837raleqdv 3247 . . . . . . 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 247 . . . . . 6 ((𝑉𝑊𝑣𝑉) → ∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
4031uvtxel 26410 . . . . . 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 701 . . . . 5 ((𝑉𝑊𝑣𝑉) → 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
4241ralrimiva 3068 . . . 4 (𝑉𝑊 → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
4311raleqdv 3247 . . . 4 (𝑉𝑊 → (∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
4442, 43mpbird 247 . . 3 (𝑉𝑊 → ∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
45 opex 5037 . . . 4 𝑉, ( I ↾ 𝑃)⟩ ∈ V
4631iscplgr 26441 . . . 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 247 . 2 (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph)
49 iscusgr 26445 . 2 (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph ↔ (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph ∧ ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph))
502, 48, 49sylanbrc 701 1 (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1596   ∈ wcel 2103   ≠ wne 2896  ∀wral 3014  ∃wrex 3015  {crab 3018  Vcvv 3304   ∖ cdif 3677   ⊆ wss 3680  𝒫 cpw 4266  {csn 4285  {cpr 4287  ⟨cop 4291   I cid 5127  ran crn 5219   ↾ cres 5220  ‘cfv 6001  (class class class)co 6765  2c2 11183  ♯chash 13232  Vtxcvtx 25994  iEdgciedg 25995  Edgcedg 26059  USGraphcusgr 26164   NeighbVtx cnbgr 26344  UnivVtxcuvtx 26406  ComplGraphccplgr 26435  ComplUSGraphccusgr 26436 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-card 8878  df-cda 9103  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-2 11192  df-n0 11406  df-z 11491  df-uz 11801  df-fz 12441  df-hash 13233  df-vtx 25996  df-iedg 25997  df-edg 26060  df-usgr 26166  df-nbgr 26345  df-uvtx 26407  df-cplgr 26437  df-cusgr 26438 This theorem is referenced by:  cusgrexg  26471
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