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Theorem cusgrasize 26006
 Description: The size of a finite complete simple graph with 𝑛 vertices (𝑛 ∈ ℕ0) is (𝑛C2) ("𝑛 choose 2") resp. (((𝑛 − 1)∗𝑛) / 2). (Contributed by Alexander van der Vekens, 11-Jan-2018.)
Assertion
Ref Expression
cusgrasize ((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin) → (#‘𝐸) = ((#‘𝑉)C2))

Proof of Theorem cusgrasize
Dummy variables 𝑒 𝑘 𝑙 𝑣 𝑓 𝑖 𝑛 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cusgra 25950 . . 3 ComplUSGrph = {⟨𝑣, 𝑒⟩ ∣ (𝑣 USGrph 𝑒 ∧ ∀𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝑒)}
21relopabi 5167 . 2 Rel ComplUSGrph
3 vex 3176 . . 3 𝑒 ∈ V
43resex 5363 . 2 (𝑒 ↾ {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)}) ∈ V
5 fveq2 6103 . . 3 (𝑒 = 𝐸 → (#‘𝑒) = (#‘𝐸))
6 fveq2 6103 . . . 4 (𝑣 = 𝑉 → (#‘𝑣) = (#‘𝑉))
76oveq1d 6564 . . 3 (𝑣 = 𝑉 → ((#‘𝑣)C2) = ((#‘𝑉)C2))
85, 7eqeqan12rd 2628 . 2 ((𝑣 = 𝑉𝑒 = 𝐸) → ((#‘𝑒) = ((#‘𝑣)C2) ↔ (#‘𝐸) = ((#‘𝑉)C2)))
9 fveq2 6103 . . 3 (𝑒 = 𝑓 → (#‘𝑒) = (#‘𝑓))
10 fveq2 6103 . . . 4 (𝑣 = 𝑤 → (#‘𝑣) = (#‘𝑤))
1110oveq1d 6564 . . 3 (𝑣 = 𝑤 → ((#‘𝑣)C2) = ((#‘𝑤)C2))
129, 11eqeqan12rd 2628 . 2 ((𝑣 = 𝑤𝑒 = 𝑓) → ((#‘𝑒) = ((#‘𝑣)C2) ↔ (#‘𝑓) = ((#‘𝑤)C2)))
13 eqid 2610 . . 3 (𝑒 ↾ {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)}) = (𝑒 ↾ {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)})
1413cusgrares 26001 . 2 ((𝑣 ComplUSGrph 𝑒𝑛𝑣) → (𝑣 ∖ {𝑛}) ComplUSGrph (𝑒 ↾ {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)}))
15 fveq2 6103 . . . 4 (𝑓 = (𝑒 ↾ {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)}) → (#‘𝑓) = (#‘(𝑒 ↾ {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)})))
1615adantl 481 . . 3 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = (𝑒 ↾ {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)})) → (#‘𝑓) = (#‘(𝑒 ↾ {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)})))
17 fveq2 6103 . . . . 5 (𝑤 = (𝑣 ∖ {𝑛}) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
1817adantr 480 . . . 4 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = (𝑒 ↾ {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)})) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
1918oveq1d 6564 . . 3 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = (𝑒 ↾ {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)})) → ((#‘𝑤)C2) = ((#‘(𝑣 ∖ {𝑛}))C2))
2016, 19eqeq12d 2625 . 2 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = (𝑒 ↾ {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)})) → ((#‘𝑓) = ((#‘𝑤)C2) ↔ (#‘(𝑒 ↾ {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)})) = ((#‘(𝑣 ∖ {𝑛}))C2)))
21 cusgrasizeindb0 25999 . 2 ((𝑣 ComplUSGrph 𝑒 ∧ (#‘𝑣) = 0) → (#‘𝑒) = ((#‘𝑣)C2))
2213cusgrasize2inds 26005 . . 3 ((𝑦 + 1) ∈ ℕ0 → ((𝑣 ComplUSGrph 𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣) → ((#‘(𝑒 ↾ {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)})) = ((#‘(𝑣 ∖ {𝑛}))C2) → (#‘𝑒) = ((#‘𝑣)C2))))
2322imp31 447 . 2 ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣 ComplUSGrph 𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ (#‘(𝑒 ↾ {𝑖 ∈ dom 𝑒𝑛 ∉ (𝑒𝑖)})) = ((#‘(𝑣 ∖ {𝑛}))C2)) → (#‘𝑒) = ((#‘𝑣)C2))
242, 4, 8, 12, 14, 20, 21, 23brfi1ind 13136 1 ((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin) → (#‘𝐸) = ((#‘𝑉)C2))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  ∀wral 2896  {crab 2900   ∖ cdif 3537  {csn 4125  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  1c1 9816   + caddc 9818  2c2 10947  ℕ0cn0 11169  Ccbc 12951  #chash 12979   USGrph cusg 25859   ComplUSGrph ccusgra 25947 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-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-usgra 25862  df-nbgra 25949  df-cusgra 25950 This theorem is referenced by:  usgramaxsize  26015
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