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Theorem fusgrfis 26144
 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 2621 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
21isfusgr 26132 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3 usgrop 25985 . . . 4 (𝐺 ∈ USGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph )
4 fvex 6168 . . . . 5 (iEdg‘𝐺) ∈ V
5 mptresid 5425 . . . . . 6 (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝} ↦ 𝑞) = ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})
6 fvex 6168 . . . . . . 7 (Edg‘⟨𝑣, 𝑒⟩) ∈ V
76mptrabex 6453 . . . . . 6 (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝} ↦ 𝑞) ∈ V
85, 7eqeltrri 2695 . . . . 5 ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ V
9 eleq1 2686 . . . . . 6 (𝑒 = (iEdg‘𝐺) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
109adantl 482 . . . . 5 ((𝑣 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
11 eleq1 2686 . . . . . 6 (𝑒 = 𝑓 → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
1211adantl 482 . . . . 5 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
13 vex 3193 . . . . . . . 8 𝑣 ∈ V
14 vex 3193 . . . . . . . 8 𝑒 ∈ V
1513, 14opvtxfvi 25823 . . . . . . 7 (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣
1615eqcomi 2630 . . . . . 6 𝑣 = (Vtx‘⟨𝑣, 𝑒⟩)
17 eqid 2621 . . . . . 6 (Edg‘⟨𝑣, 𝑒⟩) = (Edg‘⟨𝑣, 𝑒⟩)
18 eqid 2621 . . . . . 6 {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝} = {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}
19 eqid 2621 . . . . . 6 ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})⟩ = ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})⟩
2016, 17, 18, 19usgrres1 26129 . . . . 5 ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ 𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})⟩ ∈ USGraph )
21 eleq1 2686 . . . . . 6 (𝑓 = ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) → (𝑓 ∈ Fin ↔ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ Fin))
2221adantl 482 . . . . 5 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})) → (𝑓 ∈ Fin ↔ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ Fin))
2313, 14pm3.2i 471 . . . . . 6 (𝑣 ∈ V ∧ 𝑒 ∈ V)
24 fusgrfisbase 26142 . . . . . 6 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = 0) → 𝑒 ∈ Fin)
2523, 24mp3an1 1408 . . . . 5 ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = 0) → 𝑒 ∈ Fin)
26 simpl 473 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → (𝑣 ∈ V ∧ 𝑒 ∈ V))
27 simprr1 1107 . . . . . . . . . 10 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → ⟨𝑣, 𝑒⟩ ∈ USGraph )
28 eleq1 2686 . . . . . . . . . . . . . 14 ((#‘𝑣) = (𝑦 + 1) → ((#‘𝑣) ∈ ℕ0 ↔ (𝑦 + 1) ∈ ℕ0))
29 hashclb 13105 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ V → (𝑣 ∈ Fin ↔ (#‘𝑣) ∈ ℕ0))
3029biimprd 238 . . . . . . . . . . . . . . . 16 (𝑣 ∈ V → ((#‘𝑣) ∈ ℕ0𝑣 ∈ Fin))
3130adantr 481 . . . . . . . . . . . . . . 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 1081 . . . . . . . . . . . 12 ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣) → ((𝑦 + 1) ∈ ℕ0 → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin)))
3534impcom 446 . . . . . . . . . . 11 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin))
3635impcom 446 . . . . . . . . . 10 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → 𝑣 ∈ Fin)
37 opfusgr 26137 . . . . . . . . . . 11 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (⟨𝑣, 𝑒⟩ ∈ FinUSGraph ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ 𝑣 ∈ Fin)))
3837adantr 481 . . . . . . . . . 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 956 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → ⟨𝑣, 𝑒⟩ ∈ FinUSGraph )
40 simprr3 1109 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → 𝑛𝑣)
4126, 39, 403jca 1240 . . . . . . . 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 705 . . . . . . 7 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → ((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ FinUSGraph ∧ 𝑛𝑣))
43 fusgrfisstep 26143 . . . . . . 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 445 . . . . 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 13239 . . . 4 ((⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (iEdg‘𝐺) ∈ Fin)
473, 46sylan 488 . . 3 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (iEdg‘𝐺) ∈ Fin)
48 eqid 2621 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
49 eqid 2621 . . . . 5 (Edg‘𝐺) = (Edg‘𝐺)
5048, 49usgredgffibi 26138 . . . 4 (𝐺 ∈ USGraph → ((Edg‘𝐺) ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
5150adantr 481 . . 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 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ∉ wnel 2893  {crab 2912  Vcvv 3190   ∖ cdif 3557  {csn 4155  ⟨cop 4161   ↦ cmpt 4683   I cid 4994   ↾ cres 5086  ‘cfv 5857  (class class class)co 6615  Fincfn 7915  0cc0 9896  1c1 9897   + caddc 9899  ℕ0cn0 11252  #chash 13073  Vtxcvtx 25808  iEdgciedg 25809  Edgcedg 25873   USGraph cusgr 25971   FinUSGraph cfusgr 26130 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-xnn0 11324  df-z 11338  df-uz 11648  df-fz 12285  df-hash 13074  df-vtx 25810  df-iedg 25811  df-edg 25874  df-uhgr 25883  df-upgr 25907  df-umgr 25908  df-uspgr 25972  df-usgr 25973  df-fusgr 26131 This theorem is referenced by:  nbfiusgrfi  26198  cusgrsizeindslem  26268  cusgrsizeinds  26269  sizusglecusglem2  26279  vtxdgfusgrf  26313  numclwwlk1  27120
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