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Theorem cusgrsizeindslem 29469
Description: Lemma for cusgrsizeinds 29470. (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 29437 . . . . 5 (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)
2 cusgrsizeindb0.v . . . . . 6 𝑉 = (Vtx‘𝐺)
32nbcplgr 29451 . . . . 5 ((𝐺 ∈ ComplGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
41, 3sylan 580 . . . 4 ((𝐺 ∈ ComplUSGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
543adant2 1132 . . 3 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
65fveq2d 6910 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘(𝑉 ∖ {𝑁})))
7 cusgrusgr 29436 . . . . . 6 (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)
87anim1i 615 . . . . 5 ((𝐺 ∈ ComplUSGraph ∧ 𝑁𝑉) → (𝐺 ∈ USGraph ∧ 𝑁𝑉))
983adant2 1132 . . . 4 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝐺 ∈ USGraph ∧ 𝑁𝑉))
10 cusgrsizeindb0.e . . . . 5 𝐸 = (Edg‘𝐺)
112, 10nbusgrf1o 29388 . . . 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 29366 . . . . . . . 8 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
147, 13syl 17 . . . . . . 7 (𝐺 ∈ ComplUSGraph → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
1514adantr 480 . . . . . 6 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 NeighbVtx 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
16 rabfi 9303 . . . . . . 7 (𝑉 ∈ Fin → {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸} ∈ Fin)
1716adantl 481 . . . . . 6 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸} ∈ Fin)
1815, 17eqeltrd 2841 . . . . 5 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 NeighbVtx 𝑁) ∈ Fin)
19183adant3 1133 . . . 4 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) ∈ Fin)
207anim1i 615 . . . . . . 7 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
212isfusgr 29335 . . . . . . 7 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2220, 21sylibr 234 . . . . . 6 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
23 fusgrfis 29347 . . . . . . . 8 (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
2410, 23eqeltrid 2845 . . . . . . 7 (𝐺 ∈ FinUSGraph → 𝐸 ∈ Fin)
25 rabfi 9303 . . . . . . 7 (𝐸 ∈ Fin → {𝑒𝐸𝑁𝑒} ∈ Fin)
2624, 25syl 17 . . . . . 6 (𝐺 ∈ FinUSGraph → {𝑒𝐸𝑁𝑒} ∈ Fin)
2722, 26syl 17 . . . . 5 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → {𝑒𝐸𝑁𝑒} ∈ Fin)
28273adant3 1133 . . . 4 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → {𝑒𝐸𝑁𝑒} ∈ Fin)
29 hasheqf1o 14388 . . . 4 (((𝐺 NeighbVtx 𝑁) ∈ Fin ∧ {𝑒𝐸𝑁𝑒} ∈ Fin) → ((♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘{𝑒𝐸𝑁𝑒}) ↔ ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒𝐸𝑁𝑒}))
3019, 28, 29syl2anc 584 . . 3 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → ((♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘{𝑒𝐸𝑁𝑒}) ↔ ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒𝐸𝑁𝑒}))
3112, 30mpbird 257 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘{𝑒𝐸𝑁𝑒}))
32 hashdifsn 14453 . . 3 ((𝑉 ∈ Fin ∧ 𝑁𝑉) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − 1))
33323adant1 1131 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − 1))
346, 31, 333eqtr3d 2785 1 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (♯‘{𝑒𝐸𝑁𝑒}) = ((♯‘𝑉) − 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {crab 3436  cdif 3948  {csn 4626  {cpr 4628  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Fincfn 8985  1c1 11156  cmin 11492  chash 14369  Vtxcvtx 29013  Edgcedg 29064  USGraphcusgr 29166  FinUSGraphcfusgr 29333   NeighbVtx cnbgr 29349  ComplGraphccplgr 29426  ComplUSGraphccusgr 29427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370  df-vtx 29015  df-iedg 29016  df-edg 29065  df-uhgr 29075  df-upgr 29099  df-umgr 29100  df-uspgr 29167  df-usgr 29168  df-fusgr 29334  df-nbgr 29350  df-uvtx 29403  df-cplgr 29428  df-cusgr 29429
This theorem is referenced by:  cusgrsizeinds  29470
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