Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cusgrsizeindslem Structured version   Visualization version   GIF version

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-onto→wf1o 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
 Copyright terms: Public domain W3C validator