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Theorem fusgrfis 27118
Description: A finite simple graph is of finite size, i.e. has a finite number of edges. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Revised by AV, 8-Nov-2020.)
Assertion
Ref Expression
fusgrfis (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)

Proof of Theorem fusgrfis
Dummy variables 𝑒 𝑓 𝑛 𝑝 𝑞 𝑣 𝑦 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2822 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
21isfusgr 27106 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3 usgrop 26954 . . . 4 (𝐺 ∈ USGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph)
4 fvex 6665 . . . . 5 (iEdg‘𝐺) ∈ V
5 mptresid 5896 . . . . . 6 ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) = (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝} ↦ 𝑞)
6 fvex 6665 . . . . . . 7 (Edg‘⟨𝑣, 𝑒⟩) ∈ V
76mptrabex 6970 . . . . . 6 (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝} ↦ 𝑞) ∈ V
85, 7eqeltri 2910 . . . . 5 ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ V
9 eleq1 2901 . . . . . 6 (𝑒 = (iEdg‘𝐺) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
109adantl 485 . . . . 5 ((𝑣 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
11 eleq1 2901 . . . . . 6 (𝑒 = 𝑓 → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
1211adantl 485 . . . . 5 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
13 vex 3472 . . . . . . . 8 𝑣 ∈ V
14 vex 3472 . . . . . . . 8 𝑒 ∈ V
1513, 14opvtxfvi 26800 . . . . . . 7 (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣
1615eqcomi 2831 . . . . . 6 𝑣 = (Vtx‘⟨𝑣, 𝑒⟩)
17 eqid 2822 . . . . . 6 (Edg‘⟨𝑣, 𝑒⟩) = (Edg‘⟨𝑣, 𝑒⟩)
18 eqid 2822 . . . . . 6 {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝} = {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}
19 eqid 2822 . . . . . 6 ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})⟩ = ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})⟩
2016, 17, 18, 19usgrres1 27103 . . . . 5 ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ 𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})⟩ ∈ USGraph)
21 eleq1 2901 . . . . . 6 (𝑓 = ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) → (𝑓 ∈ Fin ↔ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ Fin))
2221adantl 485 . . . . 5 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})) → (𝑓 ∈ Fin ↔ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ Fin))
2313, 14pm3.2i 474 . . . . . 6 (𝑣 ∈ V ∧ 𝑒 ∈ V)
24 fusgrfisbase 27116 . . . . . 6 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = 0) → 𝑒 ∈ Fin)
2523, 24mp3an1 1445 . . . . 5 ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = 0) → 𝑒 ∈ Fin)
26 simpl 486 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → (𝑣 ∈ V ∧ 𝑒 ∈ V))
27 simprr1 1218 . . . . . . . . . 10 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → ⟨𝑣, 𝑒⟩ ∈ USGraph)
28 eleq1 2901 . . . . . . . . . . . . . 14 ((♯‘𝑣) = (𝑦 + 1) → ((♯‘𝑣) ∈ ℕ0 ↔ (𝑦 + 1) ∈ ℕ0))
29 hashclb 13715 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ V → (𝑣 ∈ Fin ↔ (♯‘𝑣) ∈ ℕ0))
3029biimprd 251 . . . . . . . . . . . . . . . 16 (𝑣 ∈ V → ((♯‘𝑣) ∈ ℕ0𝑣 ∈ Fin))
3130adantr 484 . . . . . . . . . . . . . . 15 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → ((♯‘𝑣) ∈ ℕ0𝑣 ∈ Fin))
3231com12 32 . . . . . . . . . . . . . 14 ((♯‘𝑣) ∈ ℕ0 → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin))
3328, 32syl6bir 257 . . . . . . . . . . . . 13 ((♯‘𝑣) = (𝑦 + 1) → ((𝑦 + 1) ∈ ℕ0 → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin)))
34333ad2ant2 1131 . . . . . . . . . . . 12 ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣) → ((𝑦 + 1) ∈ ℕ0 → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin)))
3534impcom 411 . . . . . . . . . . 11 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin))
3635impcom 411 . . . . . . . . . 10 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → 𝑣 ∈ Fin)
37 opfusgr 27111 . . . . . . . . . . 11 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (⟨𝑣, 𝑒⟩ ∈ FinUSGraph ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ 𝑣 ∈ Fin)))
3837adantr 484 . . . . . . . . . 10 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → (⟨𝑣, 𝑒⟩ ∈ FinUSGraph ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ 𝑣 ∈ Fin)))
3927, 36, 38mpbir2and 712 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → ⟨𝑣, 𝑒⟩ ∈ FinUSGraph)
40 simprr3 1220 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → 𝑛𝑣)
4126, 39, 403jca 1125 . . . . . . . 8 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → ((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ FinUSGraph ∧ 𝑛𝑣))
4223, 41mpan 689 . . . . . . 7 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → ((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ FinUSGraph ∧ 𝑛𝑣))
43 fusgrfisstep 27117 . . . . . . 7 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ FinUSGraph ∧ 𝑛𝑣) → (( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ Fin → 𝑒 ∈ Fin))
4442, 43syl 17 . . . . . 6 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → (( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ Fin → 𝑒 ∈ Fin))
4544imp 410 . . . . 5 ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ Fin) → 𝑒 ∈ Fin)
464, 8, 10, 12, 20, 22, 25, 45opfi1ind 13856 . . . 4 ((⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (iEdg‘𝐺) ∈ Fin)
473, 46sylan 583 . . 3 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (iEdg‘𝐺) ∈ Fin)
48 eqid 2822 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
49 eqid 2822 . . . . 5 (Edg‘𝐺) = (Edg‘𝐺)
5048, 49usgredgffibi 27112 . . . 4 (𝐺 ∈ USGraph → ((Edg‘𝐺) ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
5150adantr 484 . . 3 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → ((Edg‘𝐺) ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
5247, 51mpbird 260 . 2 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (Edg‘𝐺) ∈ Fin)
532, 52sylbi 220 1 (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wnel 3115  {crab 3134  Vcvv 3469  cdif 3905  {csn 4539  cop 4545  cmpt 5122   I cid 5436  cres 5534  cfv 6334  (class class class)co 7140  Fincfn 8496  0cc0 10526  1c1 10527   + caddc 10529  0cn0 11885  chash 13686  Vtxcvtx 26787  iEdgciedg 26788  Edgcedg 26838  USGraphcusgr 26940  FinUSGraphcfusgr 27104
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12886  df-hash 13687  df-vtx 26789  df-iedg 26790  df-edg 26839  df-uhgr 26849  df-upgr 26873  df-umgr 26874  df-uspgr 26941  df-usgr 26942  df-fusgr 27105
This theorem is referenced by:  fusgrfupgrfs  27119  nbfiusgrfi  27163  cusgrsizeindslem  27239  cusgrsizeinds  27240  sizusglecusglem2  27250  vtxdgfusgrf  27285  numclwwlk1  28144
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