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Theorem cusgrsizeindslem 40667
 Description: Lemma for cusgrsizeinds 40668. (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 40642 . . . . 5 (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)
2 cusgrsizeindb0.v . . . . . 6 𝑉 = (Vtx‘𝐺)
32nbcplgr 40656 . . . . 5 ((𝐺 ∈ ComplGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
41, 3sylan 487 . . . 4 ((𝐺 ∈ ComplUSGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
543adant2 1073 . . 3 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
65fveq2d 6107 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘(𝐺 NeighbVtx 𝑁)) = (#‘(𝑉 ∖ {𝑁})))
7 cusgrusgr 40641 . . . . . 6 (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph )
87anim1i 590 . . . . 5 ((𝐺 ∈ ComplUSGraph ∧ 𝑁𝑉) → (𝐺 ∈ USGraph ∧ 𝑁𝑉))
983adant2 1073 . . . 4 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝐺 ∈ USGraph ∧ 𝑁𝑉))
10 cusgrsizeindb0.e . . . . 5 𝐸 = (Edg‘𝐺)
112, 10nbusgrf1o 40599 . . . 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 40571 . . . . . . . 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 8070 . . . . . . 7 (𝑉 ∈ Fin → {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸} ∈ Fin)
1716adantl 481 . . . . . 6 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸} ∈ Fin)
1815, 17eqeltrd 2688 . . . . 5 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 NeighbVtx 𝑁) ∈ Fin)
19183adant3 1074 . . . 4 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) ∈ Fin)
207anim1i 590 . . . . . . 7 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
212isfusgr 40537 . . . . . . 7 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
2220, 21sylibr 223 . . . . . 6 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph )
23 fusgrfis 40549 . . . . . . . 8 (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
2410, 23syl5eqel 2692 . . . . . . 7 (𝐺 ∈ FinUSGraph → 𝐸 ∈ Fin)
25 rabfi 8070 . . . . . . 7 (𝐸 ∈ Fin → {𝑒𝐸𝑁𝑒} ∈ Fin)
2624, 25syl 17 . . . . . 6 (𝐺 ∈ FinUSGraph → {𝑒𝐸𝑁𝑒} ∈ Fin)
2722, 26syl 17 . . . . 5 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → {𝑒𝐸𝑁𝑒} ∈ Fin)
28273adant3 1074 . . . 4 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → {𝑒𝐸𝑁𝑒} ∈ Fin)
29 hasheqf1o 12999 . . . 4 (((𝐺 NeighbVtx 𝑁) ∈ Fin ∧ {𝑒𝐸𝑁𝑒} ∈ Fin) → ((#‘(𝐺 NeighbVtx 𝑁)) = (#‘{𝑒𝐸𝑁𝑒}) ↔ ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒𝐸𝑁𝑒}))
3019, 28, 29syl2anc 691 . . 3 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(𝐺 NeighbVtx 𝑁)) = (#‘{𝑒𝐸𝑁𝑒}) ↔ ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒𝐸𝑁𝑒}))
3112, 30mpbird 246 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘(𝐺 NeighbVtx 𝑁)) = (#‘{𝑒𝐸𝑁𝑒}))
32 hashdifsn 13063 . . 3 ((𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘(𝑉 ∖ {𝑁})) = ((#‘𝑉) − 1))
33323adant1 1072 . 2 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘(𝑉 ∖ {𝑁})) = ((#‘𝑉) − 1))
346, 31, 333eqtr3d 2652 1 ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘{𝑒𝐸𝑁𝑒}) = ((#‘𝑉) − 1))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {crab 2900   ∖ cdif 3537  {csn 4125  {cpr 4127  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  1c1 9816   − cmin 10145  #chash 12979  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   FinUSGraph cfusgr 40535   NeighbVtx cnbgr 40550  ComplGraphccplgr 40552  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-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  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:  cusgrsizeinds  40668
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