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Theorem cusgrsizeindslem 27237
Description: Lemma for cusgrsizeinds 27238. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)
Hypotheses
Ref Expression
cusgrsizeindb0.v 𝑉 = (Vtx‘𝐺)
cusgrsizeindb0.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
cusgrsizeindslem ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (♯‘{𝑒𝐸𝑁𝑒}) = ((♯‘𝑉) − 1))
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑉

Proof of Theorem cusgrsizeindslem
Dummy variables 𝑓 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cusgrcplgr 27206 . . . . 5 (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)
2 cusgrsizeindb0.v . . . . . 6 𝑉 = (Vtx‘𝐺)
32nbcplgr 27220 . . . . 5 ((𝐺 ∈ ComplGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
41, 3sylan 583 . . . 4 ((𝐺 ∈ ComplUSGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
543adant2 1128 . . 3 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
65fveq2d 6662 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘(𝑉 ∖ {𝑁})))
7 cusgrusgr 27205 . . . . . 6 (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)
87anim1i 617 . . . . 5 ((𝐺 ∈ ComplUSGraph ∧ 𝑁𝑉) → (𝐺 ∈ USGraph ∧ 𝑁𝑉))
983adant2 1128 . . . 4 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝐺 ∈ USGraph ∧ 𝑁𝑉))
10 cusgrsizeindb0.e . . . . 5 𝐸 = (Edg‘𝐺)
112, 10nbusgrf1o 27157 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒𝐸𝑁𝑒})
129, 11syl 17 . . 3 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒𝐸𝑁𝑒})
132, 10nbusgr 27135 . . . . . . . 8 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
147, 13syl 17 . . . . . . 7 (𝐺 ∈ ComplUSGraph → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
1514adantr 484 . . . . . 6 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
16 rabfi 8734 . . . . . . 7 (𝑉 ∈ Fin → {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸} ∈ Fin)
1716adantl 485 . . . . . 6 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸} ∈ Fin)
1815, 17eqeltrd 2916 . . . . 5 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 NeighbVtx 𝑁) ∈ Fin)
19183adant3 1129 . . . 4 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) ∈ Fin)
207anim1i 617 . . . . . . 7 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
212isfusgr 27104 . . . . . . 7 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2220, 21sylibr 237 . . . . . 6 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
23 fusgrfis 27116 . . . . . . . 8 (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
2410, 23eqeltrid 2920 . . . . . . 7 (𝐺 ∈ FinUSGraph → 𝐸 ∈ Fin)
25 rabfi 8734 . . . . . . 7 (𝐸 ∈ Fin → {𝑒𝐸𝑁𝑒} ∈ Fin)
2624, 25syl 17 . . . . . 6 (𝐺 ∈ FinUSGraph → {𝑒𝐸𝑁𝑒} ∈ Fin)
2722, 26syl 17 . . . . 5 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → {𝑒𝐸𝑁𝑒} ∈ Fin)
28273adant3 1129 . . . 4 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → {𝑒𝐸𝑁𝑒} ∈ Fin)
29 hasheqf1o 13710 . . . 4 (((𝐺 NeighbVtx 𝑁) ∈ Fin ∧ {𝑒𝐸𝑁𝑒} ∈ Fin) → ((♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘{𝑒𝐸𝑁𝑒}) ↔ ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒𝐸𝑁𝑒}))
3019, 28, 29syl2anc 587 . . 3 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → ((♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘{𝑒𝐸𝑁𝑒}) ↔ ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒𝐸𝑁𝑒}))
3112, 30mpbird 260 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘{𝑒𝐸𝑁𝑒}))
32 hashdifsn 13776 . . 3 ((𝑉 ∈ Fin ∧ 𝑁𝑉) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − 1))
33323adant1 1127 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − 1))
346, 31, 333eqtr3d 2867 1 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (♯‘{𝑒𝐸𝑁𝑒}) = ((♯‘𝑉) − 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  {crab 3137  cdif 3916  {csn 4549  {cpr 4551  1-1-ontowf1o 6342  cfv 6343  (class class class)co 7145  Fincfn 8499  1c1 10530  cmin 10862  chash 13691  Vtxcvtx 26785  Edgcedg 26836  USGraphcusgr 26938  FinUSGraphcfusgr 27102   NeighbVtx cnbgr 27118  ComplGraphccplgr 27195  ComplUSGraphccusgr 27196
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7571  df-1st 7679  df-2nd 7680  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-1o 8092  df-2o 8093  df-oadd 8096  df-er 8279  df-en 8500  df-dom 8501  df-sdom 8502  df-fin 8503  df-dju 9321  df-card 9359  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11693  df-n0 11891  df-xnn0 11961  df-z 11975  df-uz 12237  df-fz 12891  df-hash 13692  df-vtx 26787  df-iedg 26788  df-edg 26837  df-uhgr 26847  df-upgr 26871  df-umgr 26872  df-uspgr 26939  df-usgr 26940  df-fusgr 27103  df-nbgr 27119  df-uvtx 27172  df-cplgr 27197  df-cusgr 27198
This theorem is referenced by:  cusgrsizeinds  27238
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