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Theorem cusgrsize 40670
 Description: The size of a finite complete simple graph with 𝑛 vertices (𝑛 ∈ ℕ0) is (𝑛C2) ("𝑛 choose 2") resp. (((𝑛 − 1)∗𝑛) / 2), see definition in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 10-Nov-2020.)
Hypotheses
Ref Expression
cusgrsizeindb0.v 𝑉 = (Vtx‘𝐺)
cusgrsizeindb0.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
cusgrsize ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘𝐸) = ((#‘𝑉)C2))

Proof of Theorem cusgrsize
Dummy variables 𝑒 𝑓 𝑛 𝑣 𝑐 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cusgrsizeindb0.e . . . . 5 𝐸 = (Edg‘𝐺)
2 edgaval 25794 . . . . 5 (𝐺 ∈ ComplUSGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
31, 2syl5eq 2656 . . . 4 (𝐺 ∈ ComplUSGraph → 𝐸 = ran (iEdg‘𝐺))
43adantr 480 . . 3 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐸 = ran (iEdg‘𝐺))
54fveq2d 6107 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘𝐸) = (#‘ran (iEdg‘𝐺)))
6 cusgrsizeindb0.v . . . . 5 𝑉 = (Vtx‘𝐺)
76opeq1i 4343 . . . 4 𝑉, (iEdg‘𝐺)⟩ = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩
8 cusgrop 40660 . . . 4 (𝐺 ∈ ComplUSGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplUSGraph)
97, 8syl5eqel 2692 . . 3 (𝐺 ∈ ComplUSGraph → ⟨𝑉, (iEdg‘𝐺)⟩ ∈ ComplUSGraph)
10 fvex 6113 . . . 4 (iEdg‘𝐺) ∈ V
11 fvex 6113 . . . . 5 (Edg‘⟨𝑣, 𝑒⟩) ∈ V
12 rabexg 4739 . . . . . 6 ((Edg‘⟨𝑣, 𝑒⟩) ∈ V → {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐} ∈ V)
1312resiexd 6385 . . . . 5 ((Edg‘⟨𝑣, 𝑒⟩) ∈ V → ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐}) ∈ V)
1411, 13ax-mp 5 . . . 4 ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐}) ∈ V
15 rneq 5272 . . . . . 6 (𝑒 = (iEdg‘𝐺) → ran 𝑒 = ran (iEdg‘𝐺))
1615fveq2d 6107 . . . . 5 (𝑒 = (iEdg‘𝐺) → (#‘ran 𝑒) = (#‘ran (iEdg‘𝐺)))
17 fveq2 6103 . . . . . 6 (𝑣 = 𝑉 → (#‘𝑣) = (#‘𝑉))
1817oveq1d 6564 . . . . 5 (𝑣 = 𝑉 → ((#‘𝑣)C2) = ((#‘𝑉)C2))
1916, 18eqeqan12rd 2628 . . . 4 ((𝑣 = 𝑉𝑒 = (iEdg‘𝐺)) → ((#‘ran 𝑒) = ((#‘𝑣)C2) ↔ (#‘ran (iEdg‘𝐺)) = ((#‘𝑉)C2)))
20 rneq 5272 . . . . . 6 (𝑒 = 𝑓 → ran 𝑒 = ran 𝑓)
2120fveq2d 6107 . . . . 5 (𝑒 = 𝑓 → (#‘ran 𝑒) = (#‘ran 𝑓))
22 fveq2 6103 . . . . . 6 (𝑣 = 𝑤 → (#‘𝑣) = (#‘𝑤))
2322oveq1d 6564 . . . . 5 (𝑣 = 𝑤 → ((#‘𝑣)C2) = ((#‘𝑤)C2))
2421, 23eqeqan12rd 2628 . . . 4 ((𝑣 = 𝑤𝑒 = 𝑓) → ((#‘ran 𝑒) = ((#‘𝑣)C2) ↔ (#‘ran 𝑓) = ((#‘𝑤)C2)))
25 vex 3176 . . . . . . 7 𝑣 ∈ V
26 vex 3176 . . . . . . 7 𝑒 ∈ V
27 opvtxfv 25681 . . . . . . 7 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣)
2825, 26, 27mp2an 704 . . . . . 6 (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣
2928eqcomi 2619 . . . . 5 𝑣 = (Vtx‘⟨𝑣, 𝑒⟩)
30 eqid 2610 . . . . 5 (Edg‘⟨𝑣, 𝑒⟩) = (Edg‘⟨𝑣, 𝑒⟩)
31 eqid 2610 . . . . 5 {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐} = {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐}
32 eqid 2610 . . . . 5 ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})⟩ = ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})⟩
3329, 30, 31, 32cusgrres 40664 . . . 4 ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ 𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})⟩ ∈ ComplUSGraph)
34 rneq 5272 . . . . . . 7 (𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐}) → ran 𝑓 = ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐}))
3534fveq2d 6107 . . . . . 6 (𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐}) → (#‘ran 𝑓) = (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})))
3635adantl 481 . . . . 5 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})) → (#‘ran 𝑓) = (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})))
37 fveq2 6103 . . . . . . 7 (𝑤 = (𝑣 ∖ {𝑛}) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
3837adantr 480 . . . . . 6 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
3938oveq1d 6564 . . . . 5 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})) → ((#‘𝑤)C2) = ((#‘(𝑣 ∖ {𝑛}))C2))
4036, 39eqeq12d 2625 . . . 4 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})) → ((#‘ran 𝑓) = ((#‘𝑤)C2) ↔ (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})) = ((#‘(𝑣 ∖ {𝑛}))C2)))
41 edgaopval 25795 . . . . . . . . 9 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
4225, 26, 41mp2an 704 . . . . . . . 8 (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒
4342a1i 11 . . . . . . 7 ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = 0) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
4443eqcomd 2616 . . . . . 6 ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = 0) → ran 𝑒 = (Edg‘⟨𝑣, 𝑒⟩))
4544fveq2d 6107 . . . . 5 ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = 0) → (#‘ran 𝑒) = (#‘(Edg‘⟨𝑣, 𝑒⟩)))
46 cusgrusgr 40641 . . . . . . 7 (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph → ⟨𝑣, 𝑒⟩ ∈ USGraph )
47 usgruhgr 40413 . . . . . . 7 (⟨𝑣, 𝑒⟩ ∈ USGraph → ⟨𝑣, 𝑒⟩ ∈ UHGraph )
4846, 47syl 17 . . . . . 6 (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph → ⟨𝑣, 𝑒⟩ ∈ UHGraph )
4929, 30cusgrsizeindb0 40665 . . . . . 6 ((⟨𝑣, 𝑒⟩ ∈ UHGraph ∧ (#‘𝑣) = 0) → (#‘(Edg‘⟨𝑣, 𝑒⟩)) = ((#‘𝑣)C2))
5048, 49sylan 487 . . . . 5 ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = 0) → (#‘(Edg‘⟨𝑣, 𝑒⟩)) = ((#‘𝑣)C2))
5145, 50eqtrd 2644 . . . 4 ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = 0) → (#‘ran 𝑒) = ((#‘𝑣)C2))
52 rnresi 5398 . . . . . . . . . 10 ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐}) = {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐}
5352fveq2i 6106 . . . . . . . . 9 (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})) = (#‘{𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})
5442a1i 11 . . . . . . . . . . 11 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
5554rabeqdv 3167 . . . . . . . . . 10 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐} = {𝑐 ∈ ran 𝑒𝑛𝑐})
5655fveq2d 6107 . . . . . . . . 9 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → (#‘{𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐}) = (#‘{𝑐 ∈ ran 𝑒𝑛𝑐}))
5753, 56syl5eq 2656 . . . . . . . 8 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})) = (#‘{𝑐 ∈ ran 𝑒𝑛𝑐}))
5857eqeq1d 2612 . . . . . . 7 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → ((#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})) = ((#‘(𝑣 ∖ {𝑛}))C2) ↔ (#‘{𝑐 ∈ ran 𝑒𝑛𝑐}) = ((#‘(𝑣 ∖ {𝑛}))C2)))
5958biimpd 218 . . . . . 6 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → ((#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})) = ((#‘(𝑣 ∖ {𝑛}))C2) → (#‘{𝑐 ∈ ran 𝑒𝑛𝑐}) = ((#‘(𝑣 ∖ {𝑛}))C2)))
6059imdistani 722 . . . . 5 ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})) = ((#‘(𝑣 ∖ {𝑛}))C2)) → (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ (#‘{𝑐 ∈ ran 𝑒𝑛𝑐}) = ((#‘(𝑣 ∖ {𝑛}))C2)))
6142eqcomi 2619 . . . . . . 7 ran 𝑒 = (Edg‘⟨𝑣, 𝑒⟩)
62 eqid 2610 . . . . . . 7 {𝑐 ∈ ran 𝑒𝑛𝑐} = {𝑐 ∈ ran 𝑒𝑛𝑐}
6329, 61, 62cusgrsize2inds 40669 . . . . . 6 ((𝑦 + 1) ∈ ℕ0 → ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣) → ((#‘{𝑐 ∈ ran 𝑒𝑛𝑐}) = ((#‘(𝑣 ∖ {𝑛}))C2) → (#‘ran 𝑒) = ((#‘𝑣)C2))))
6463imp31 447 . . . . 5 ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ (#‘{𝑐 ∈ ran 𝑒𝑛𝑐}) = ((#‘(𝑣 ∖ {𝑛}))C2)) → (#‘ran 𝑒) = ((#‘𝑣)C2))
6560, 64syl 17 . . . 4 ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑐})) = ((#‘(𝑣 ∖ {𝑛}))C2)) → (#‘ran 𝑒) = ((#‘𝑣)C2))
6610, 14, 19, 24, 33, 40, 51, 65opfi1ind 13139 . . 3 ((⟨𝑉, (iEdg‘𝐺)⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘ran (iEdg‘𝐺)) = ((#‘𝑉)C2))
679, 66sylan 487 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘ran (iEdg‘𝐺)) = ((#‘𝑉)C2))
685, 67eqtrd 2644 1 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘𝐸) = ((#‘𝑉)C2))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  {crab 2900  Vcvv 3173   ∖ cdif 3537  {csn 4125  ⟨cop 4131   I cid 4948  ran crn 5039   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  ℕ0cn0 11169  Ccbc 12951  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722  Edgcedga 25792   USGraph cusgr 40379  ComplUSGraphccusgr 40553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-seq 12664  df-fac 12923  df-bc 12952  df-hash 12980  df-vtx 25675  df-iedg 25676  df-uhgr 25724  df-upgr 25749  df-umgr 25750  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-fusgr 40536  df-nbgr 40554  df-uvtxa 40556  df-cplgr 40557  df-cusgr 40558 This theorem is referenced by:  fusgrmaxsize  40680
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