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Theorem fusgrfis 26445
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 2771 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
21isfusgr 26433 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3 usgrop 26280 . . . 4 (𝐺 ∈ USGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph)
4 fvex 6344 . . . . 5 (iEdg‘𝐺) ∈ V
5 mptresid 5596 . . . . . 6 (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝} ↦ 𝑞) = ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})
6 fvex 6344 . . . . . . 7 (Edg‘⟨𝑣, 𝑒⟩) ∈ V
76mptrabex 6635 . . . . . 6 (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝} ↦ 𝑞) ∈ V
85, 7eqeltrri 2847 . . . . 5 ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ V
9 eleq1 2838 . . . . . 6 (𝑒 = (iEdg‘𝐺) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
109adantl 467 . . . . 5 ((𝑣 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
11 eleq1 2838 . . . . . 6 (𝑒 = 𝑓 → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
1211adantl 467 . . . . 5 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
13 vex 3354 . . . . . . . 8 𝑣 ∈ V
14 vex 3354 . . . . . . . 8 𝑒 ∈ V
1513, 14opvtxfvi 26110 . . . . . . 7 (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣
1615eqcomi 2780 . . . . . 6 𝑣 = (Vtx‘⟨𝑣, 𝑒⟩)
17 eqid 2771 . . . . . 6 (Edg‘⟨𝑣, 𝑒⟩) = (Edg‘⟨𝑣, 𝑒⟩)
18 eqid 2771 . . . . . 6 {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝} = {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}
19 eqid 2771 . . . . . 6 ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})⟩ = ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})⟩
2016, 17, 18, 19usgrres1 26430 . . . . 5 ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ 𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})⟩ ∈ USGraph)
21 eleq1 2838 . . . . . 6 (𝑓 = ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) → (𝑓 ∈ Fin ↔ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ Fin))
2221adantl 467 . . . . 5 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})) → (𝑓 ∈ Fin ↔ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ Fin))
2313, 14pm3.2i 456 . . . . . 6 (𝑣 ∈ V ∧ 𝑒 ∈ V)
24 fusgrfisbase 26443 . . . . . 6 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = 0) → 𝑒 ∈ Fin)
2523, 24mp3an1 1559 . . . . 5 ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = 0) → 𝑒 ∈ Fin)
26 simpl 468 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → (𝑣 ∈ V ∧ 𝑒 ∈ V))
27 simprr1 1272 . . . . . . . . . 10 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → ⟨𝑣, 𝑒⟩ ∈ USGraph)
28 eleq1 2838 . . . . . . . . . . . . . 14 ((♯‘𝑣) = (𝑦 + 1) → ((♯‘𝑣) ∈ ℕ0 ↔ (𝑦 + 1) ∈ ℕ0))
29 hashclb 13351 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ V → (𝑣 ∈ Fin ↔ (♯‘𝑣) ∈ ℕ0))
3029biimprd 238 . . . . . . . . . . . . . . . 16 (𝑣 ∈ V → ((♯‘𝑣) ∈ ℕ0𝑣 ∈ Fin))
3130adantr 466 . . . . . . . . . . . . . . 15 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → ((♯‘𝑣) ∈ ℕ0𝑣 ∈ Fin))
3231com12 32 . . . . . . . . . . . . . 14 ((♯‘𝑣) ∈ ℕ0 → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin))
3328, 32syl6bir 244 . . . . . . . . . . . . 13 ((♯‘𝑣) = (𝑦 + 1) → ((𝑦 + 1) ∈ ℕ0 → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin)))
34333ad2ant2 1128 . . . . . . . . . . . 12 ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣) → ((𝑦 + 1) ∈ ℕ0 → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin)))
3534impcom 394 . . . . . . . . . . 11 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin))
3635impcom 394 . . . . . . . . . 10 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → 𝑣 ∈ Fin)
37 opfusgr 26438 . . . . . . . . . . 11 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (⟨𝑣, 𝑒⟩ ∈ FinUSGraph ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ 𝑣 ∈ Fin)))
3837adantr 466 . . . . . . . . . 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 692 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → ⟨𝑣, 𝑒⟩ ∈ FinUSGraph)
40 simprr3 1276 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → 𝑛𝑣)
4126, 39, 403jca 1122 . . . . . . . 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 670 . . . . . . 7 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → ((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ FinUSGraph ∧ 𝑛𝑣))
43 fusgrfisstep 26444 . . . . . . 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 393 . . . . 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 13486 . . . 4 ((⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (iEdg‘𝐺) ∈ Fin)
473, 46sylan 569 . . 3 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (iEdg‘𝐺) ∈ Fin)
48 eqid 2771 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
49 eqid 2771 . . . . 5 (Edg‘𝐺) = (Edg‘𝐺)
5048, 49usgredgffibi 26439 . . . 4 (𝐺 ∈ USGraph → ((Edg‘𝐺) ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
5150adantr 466 . . 3 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → ((Edg‘𝐺) ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
5247, 51mpbird 247 . 2 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (Edg‘𝐺) ∈ Fin)
532, 52sylbi 207 1 (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wnel 3046  {crab 3065  Vcvv 3351  cdif 3720  {csn 4317  cop 4323  cmpt 4864   I cid 5157  cres 5252  cfv 6030  (class class class)co 6796  Fincfn 8113  0cc0 10142  1c1 10143   + caddc 10145  0cn0 11499  chash 13321  Vtxcvtx 26095  iEdgciedg 26096  Edgcedg 26160  USGraphcusgr 26266  FinUSGraphcfusgr 26431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8969  df-cda 9196  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-n0 11500  df-xnn0 11571  df-z 11585  df-uz 11894  df-fz 12534  df-hash 13322  df-vtx 26097  df-iedg 26098  df-edg 26161  df-uhgr 26174  df-upgr 26198  df-umgr 26199  df-uspgr 26267  df-usgr 26268  df-fusgr 26432
This theorem is referenced by:  fusgrfupgrfs  26446  nbfiusgrfi  26500  cusgrsizeindslem  26582  cusgrsizeinds  26583  sizusglecusglem2  26593  vtxdgfusgrf  26628  numclwwlk1  27548
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