Theorem cusgrexi 29499

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 29497	. 2 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph)
3	1	cusgrexilem1 29495	. . . . . . . . 9 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃) ∈ V)
4		opvtxfv 29060	. . . . . . . . . 10 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
5	4	eqcomd 2743	. . . . . . . . 9 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑃) ∈ V) → 𝑉 = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
6	3, 5	mpdan 688	. . . . . . . 8 ⊢ (𝑉 ∈ 𝑊 → 𝑉 = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
7	6	eleq2d 2823	. . . . . . 7 ⊢ (𝑉 ∈ 𝑊 → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
8	7	biimpa 476	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
9		eldifi 4084	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛 ∈ 𝑉)
10	9	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ 𝑉)
11	3, 4	mpdan 688	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
12	11	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝑊 → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑛 ∈ 𝑉))
13	12	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑛 ∈ 𝑉))
14	10, 13	mpbird 257	. . . . . . . . . 10 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
15		simplr 769	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑣 ∈ 𝑉)
16	11	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝑊 → (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑣 ∈ 𝑉))
17	16	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ 𝑣 ∈ 𝑉))
18	15, 17	mpbird 257	. . . . . . . . . 10 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
19	14, 18	jca 511	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → (𝑛 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∧ 𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
20		eldifsni 4747	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝑉 ∖ {𝑣}) → 𝑛 ≠ 𝑣)
21	20	adantl 481	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ≠ 𝑣)
22	1	cusgrexilem2 29498	. . . . . . . . . 10 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒)
23		edgval 29105	. . . . . . . . . . . . 13 ⊢ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩)
24		opiedgfv 29063	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
25	3, 24	mpdan 688	. . . . . . . . . . . . . 14 ⊢ (𝑉 ∈ 𝑊 → (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ( I ↾ 𝑃))
26	25	rneqd 5888	. . . . . . . . . . . . 13 ⊢ (𝑉 ∈ 𝑊 → ran (iEdg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran ( I ↾ 𝑃))
27	23, 26	eqtrid 2784	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝑊 → (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = ran ( I ↾ 𝑃))
28	27	rexeqdv 3298	. . . . . . . . . . 11 ⊢ (𝑉 ∈ 𝑊 → (∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒))
29	28	ad2antrr 727	. . . . . . . . . 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 257	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩){𝑣, 𝑛} ⊆ 𝑒)
31		eqid 2737	. . . . . . . . . 10 ⊢ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)
32		eqid 2737	. . . . . . . . . 10 ⊢ (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩) = (Edg‘⟨𝑉, ( I ↾ 𝑃)⟩)
33	31, 32	nbgrel 29396	. . . . . . . . 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 1345	. . . . . . . 8 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → 𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
35	34	ralrimiva 3129	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
36	11	adantr 480	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) = 𝑉)
37	36	difeq1d 4078	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣}) = (𝑉 ∖ {𝑣}))
38	35, 37	raleqtrrdv 3301	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → ∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣))
39	31	uvtxel 29444	. . . . . 6 ⊢ (𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ↔ (𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∧ ∀𝑛 ∈ ((Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩) ∖ {𝑣})𝑛 ∈ (⟨𝑉, ( I ↾ 𝑃)⟩ NeighbVtx 𝑣)))
40	8, 38, 39	sylanbrc 584	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
41	40	ralrimiva 3129	. . . 4 ⊢ (𝑉 ∈ 𝑊 → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
42	41, 11	raleqtrrdv 3301	. . 3 ⊢ (𝑉 ∈ 𝑊 → ∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩))
43		opex 5413	. . . 4 ⊢ ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ V
44	31	iscplgr 29471	. . . 4 ⊢ (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ V → (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
45	43, 44	mp1i 13	. . 3 ⊢ (𝑉 ∈ 𝑊 → (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph ↔ ∀𝑣 ∈ (Vtx‘⟨𝑉, ( I ↾ 𝑃)⟩)𝑣 ∈ (UnivVtx‘⟨𝑉, ( I ↾ 𝑃)⟩)))
46	42, 45	mpbird 257	. 2 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph)
47		iscusgr 29474	. 2 ⊢ (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph ↔ (⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph ∧ ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplGraph))
48	2, 46, 47	sylanbrc 584	1 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  {crab 3400  Vcvv 3441   ∖ cdif 3899   ⊆ wss 3902  𝒫 cpw 4555  {csn 4581  {cpr 4583  ⟨cop 4587   I cid 5519  ran crn 5626   ↾ cres 5627  ‘cfv 6493  (class class class)co 7360  2c2 12204  ♯chash 14257  Vtxcvtx 29052  iEdgciedg 29053  Edgcedg 29103  USGraphcusgr 29205   NeighbVtx cnbgr 29388  UnivVtxcuvtx 29441  ComplGraphccplgr 29465  ComplUSGraphccusgr 29466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-oadd 8403  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9817  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-n0 12406  df-z 12493  df-uz 12756  df-fz 13428  df-hash 14258  df-vtx 29054  df-iedg 29055  df-edg 29104  df-usgr 29207  df-nbgr 29389  df-uvtx 29442  df-cplgr 29467  df-cusgr 29468

This theorem is referenced by:  cusgrexg  29500