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Theorem fusgrfis 40549
 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 2610 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
21isfusgr 40537 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3 usgrop 40393 . . . 4 (𝐺 ∈ USGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph )
4 fvex 6113 . . . . 5 (iEdg‘𝐺) ∈ V
5 mptresid 5375 . . . . . 6 (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝} ↦ 𝑞) = ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})
6 fvex 6113 . . . . . . 7 (Edg‘⟨𝑣, 𝑒⟩) ∈ V
76mptrabex 6392 . . . . . 6 (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝} ↦ 𝑞) ∈ V
85, 7eqeltrri 2685 . . . . 5 ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}) ∈ V
9 eleq1 2676 . . . . . 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 2676 . . . . . 6 (𝑒 = 𝑓 → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
1211adantl 481 . . . . 5 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
13 vex 3176 . . . . . . . 8 𝑣 ∈ V
14 vex 3176 . . . . . . . 8 𝑒 ∈ V
1513, 14opvtxfvi 25686 . . . . . . 7 (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣
1615eqcomi 2619 . . . . . 6 𝑣 = (Vtx‘⟨𝑣, 𝑒⟩)
17 eqid 2610 . . . . . 6 (Edg‘⟨𝑣, 𝑒⟩) = (Edg‘⟨𝑣, 𝑒⟩)
18 eqid 2610 . . . . . 6 {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝} = {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝}
19 eqid 2610 . . . . . 6 ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})⟩ = ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})⟩
2016, 17, 18, 19usgrres1 40534 . . . . 5 ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ 𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛𝑝})⟩ ∈ USGraph )
21 eleq1 2676 . . . . . 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 40547 . . . . . 6 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = 0) → 𝑒 ∈ Fin)
2523, 24mp3an1 1403 . . . . 5 ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = 0) → 𝑒 ∈ Fin)
26 simpl 472 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → (𝑣 ∈ V ∧ 𝑒 ∈ V))
27 simprr1 1102 . . . . . . . . . 10 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → ⟨𝑣, 𝑒⟩ ∈ USGraph )
28 eleq1 2676 . . . . . . . . . . . . . 14 ((#‘𝑣) = (𝑦 + 1) → ((#‘𝑣) ∈ ℕ0 ↔ (𝑦 + 1) ∈ ℕ0))
29 hashclb 13011 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ V → (𝑣 ∈ Fin ↔ (#‘𝑣) ∈ ℕ0))
3029biimprd 237 . . . . . . . . . . . . . . . 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, 32syl6bir 243 . . . . . . . . . . . . 13 ((#‘𝑣) = (𝑦 + 1) → ((𝑦 + 1) ∈ ℕ0 → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin)))
34333ad2ant2 1076 . . . . . . . . . . . 12 ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣) → ((𝑦 + 1) ∈ ℕ0 → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin)))
3534impcom 445 . . . . . . . . . . 11 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin))
3635impcom 445 . . . . . . . . . 10 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → 𝑣 ∈ Fin)
37 opfusgr 40542 . . . . . . . . . . 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 959 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → ⟨𝑣, 𝑒⟩ ∈ FinUSGraph )
40 simprr3 1104 . . . . . . . . 9 (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))) → 𝑛𝑣)
4126, 39, 403jca 1235 . . . . . . . 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 702 . . . . . . 7 (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) → ((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ FinUSGraph ∧ 𝑛𝑣))
43 fusgrfisstep 40548 . . . . . . 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 444 . . . . 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 13139 . . . 4 ((⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (iEdg‘𝐺) ∈ Fin)
473, 46sylan 487 . . 3 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (iEdg‘𝐺) ∈ Fin)
48 eqid 2610 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
49 eqid 2610 . . . . 5 (Edg‘𝐺) = (Edg‘𝐺)
5048, 49usgredgffibi 40543 . . . 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 246 . 2 ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (Edg‘𝐺) ∈ Fin)
532, 52sylbi 206 1 (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  {crab 2900  Vcvv 3173   ∖ cdif 3537  {csn 4125  ⟨cop 4131   ↦ cmpt 4643   I cid 4948   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  1c1 9816   + caddc 9818  ℕ0cn0 11169  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674  Edgcedga 25792   USGraph cusgr 40379   FinUSGraph cfusgr 40535 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-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 This theorem is referenced by:  nbfiusgrfi  40603  cusgrsizeindslem  40667  cusgrsizeinds  40668  sizusglecusglem2  40678  vtxdgfusgrf  40712  av-numclwwlk1  41528
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