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Theorem fusgrfis 29361
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 2734 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
21isfusgr 29349 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3 usgrop 29194 . . . 4 (𝐺 ∈ USGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph)
4 fvex 6919 . . . . 5 (iEdg‘𝐺) ∈ V
5 mptresid 6070 . . . . . 6 ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) = (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝} ↦ 𝑞)
6 fvex 6919 . . . . . . 7 (Edg‘⟨𝑣, 𝑒⟩) ∈ V
76mptrabex 7244 . . . . . 6 (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝} ↦ 𝑞) ∈ V
85, 7eqeltri 2834 . . . . 5 ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ V
9 eleq1 2826 . . . . . 6 (𝑒 = (iEdg‘𝐺) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
109adantl 481 . . . . 5 ((𝑣 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
11 eleq1 2826 . . . . . 6 (𝑒 = 𝑓 → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
1211adantl 481 . . . . 5 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
13 vex 3481 . . . . . . . 8 𝑣 ∈ V
14 vex 3481 . . . . . . . 8 𝑒 ∈ V
1513, 14opvtxfvi 29040 . . . . . . 7 (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣
1615eqcomi 2743 . . . . . 6 𝑣 = (Vtx‘⟨𝑣, 𝑒⟩)
17 eqid 2734 . . . . . 6 (Edg‘⟨𝑣, 𝑒⟩) = (Edg‘⟨𝑣, 𝑒⟩)
18 eqid 2734 . . . . . 6 {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝} = {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}
19 eqid 2734 . . . . . 6 ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})⟩ = ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})⟩
2016, 17, 18, 19usgrres1 29346 . . . . 5 ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ 𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})⟩ ∈ USGraph)
21 eleq1 2826 . . . . . 6 (𝑓 = ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) → (𝑓 ∈ Fin ↔ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ Fin))
2221adantl 481 . . . . 5 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})) → (𝑓 ∈ Fin ↔ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ Fin))
2313, 14pm3.2i 470 . . . . . 6 (𝑣 ∈ V ∧ 𝑒 ∈ V)
24 fusgrfisbase 29359 . . . . . 6 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = 0) → 𝑒 ∈ Fin)
2523, 24mp3an1 1447 . . . . 5 ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = 0) → 𝑒 ∈ Fin)
26 simpl 482 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → (𝑣 ∈ V ∧ 𝑒 ∈ V))
27 simprr1 1220 . . . . . . . . . 10 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → ⟨𝑣, 𝑒⟩ ∈ USGraph)
28 eleq1 2826 . . . . . . . . . . . . . 14 ((♯‘𝑣) = (𝑦 + 1) → ((♯‘𝑣) ∈ ℕ0 ↔ (𝑦 + 1) ∈ ℕ0))
29 hashclb 14393 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ V → (𝑣 ∈ Fin ↔ (♯‘𝑣) ∈ ℕ0))
3029biimprd 248 . . . . . . . . . . . . . . . 16 (𝑣 ∈ V → ((♯‘𝑣) ∈ ℕ0𝑣 ∈ Fin))
3130adantr 480 . . . . . . . . . . . . . . 15 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → ((♯‘𝑣) ∈ ℕ0𝑣 ∈ Fin))
3231com12 32 . . . . . . . . . . . . . 14 ((♯‘𝑣) ∈ ℕ0 → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin))
3328, 32biimtrrdi 254 . . . . . . . . . . . . 13 ((♯‘𝑣) = (𝑦 + 1) → ((𝑦 + 1) ∈ ℕ0 → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin)))
34333ad2ant2 1133 . . . . . . . . . . . 12 ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣) → ((𝑦 + 1) ∈ ℕ0 → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin)))
3534impcom 407 . . . . . . . . . . 11 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin))
3635impcom 407 . . . . . . . . . 10 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → 𝑣 ∈ Fin)
37 opfusgr 29354 . . . . . . . . . . 11 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (⟨𝑣, 𝑒⟩ ∈ FinUSGraph ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ 𝑣 ∈ Fin)))
3837adantr 480 . . . . . . . . . 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 713 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → ⟨𝑣, 𝑒⟩ ∈ FinUSGraph)
40 simprr3 1222 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → 𝑛𝑣)
4126, 39, 403jca 1127 . . . . . . . 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 690 . . . . . . 7 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → ((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ FinUSGraph ∧ 𝑛𝑣))
43 fusgrfisstep 29360 . . . . . . 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 406 . . . . 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 14547 . . . 4 ((⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (iEdg‘𝐺) ∈ Fin)
473, 46sylan 580 . . 3 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (iEdg‘𝐺) ∈ Fin)
48 eqid 2734 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
49 eqid 2734 . . . . 5 (Edg‘𝐺) = (Edg‘𝐺)
5048, 49usgredgffibi 29355 . . . 4 (𝐺 ∈ USGraph → ((Edg‘𝐺) ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
5150adantr 480 . . 3 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → ((Edg‘𝐺) ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
5247, 51mpbird 257 . 2 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (Edg‘𝐺) ∈ Fin)
532, 52sylbi 217 1 (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wnel 3043  {crab 3432  Vcvv 3477  cdif 3959  {csn 4630  cop 4636  cmpt 5230   I cid 5581  cres 5690  cfv 6562  (class class class)co 7430  Fincfn 8983  0cc0 11152  1c1 11153   + caddc 11155  0cn0 12523  chash 14365  Vtxcvtx 29027  iEdgciedg 29028  Edgcedg 29078  USGraphcusgr 29180  FinUSGraphcfusgr 29347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-hash 14366  df-vtx 29029  df-iedg 29030  df-edg 29079  df-uhgr 29089  df-upgr 29113  df-umgr 29114  df-uspgr 29181  df-usgr 29182  df-fusgr 29348
This theorem is referenced by:  fusgrfupgrfs  29362  nbfiusgrfi  29406  cusgrsizeindslem  29483  cusgrsizeinds  29484  sizusglecusglem2  29494  vtxdgfusgrf  29529  numclwwlk1  30389  clnbfiusgrfi  47767
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