Theorem fusgrfis 29264

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.

Step	Hyp	Ref	Expression
1		eqid 2730	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29252	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3		usgrop 29097	. . . 4 ⊢ (𝐺 ∈ USGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph)
4		fvex 6874	. . . . 5 ⊢ (iEdg‘𝐺) ∈ V
5		mptresid 6025	. . . . . 6 ⊢ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝}) = (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝} ↦ 𝑞)
6		fvex 6874	. . . . . . 7 ⊢ (Edg‘⟨𝑣, 𝑒⟩) ∈ V
7	6	mptrabex 7202	. . . . . 6 ⊢ (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝} ↦ 𝑞) ∈ V
8	5, 7	eqeltri 2825	. . . . 5 ⊢ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝}) ∈ V
9		eleq1 2817	. . . . . 6 ⊢ (𝑒 = (iEdg‘𝐺) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
10	9	adantl 481	. . . . 5 ⊢ ((𝑣 = (Vtx‘𝐺) ∧ 𝑒 = (iEdg‘𝐺)) → (𝑒 ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
11		eleq1 2817	. . . . . 6 ⊢ (𝑒 = 𝑓 → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
12	11	adantl 481	. . . . 5 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
13		vex 3454	. . . . . . . 8 ⊢ 𝑣 ∈ V
14		vex 3454	. . . . . . . 8 ⊢ 𝑒 ∈ V
15	13, 14	opvtxfvi 28943	. . . . . . 7 ⊢ (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣
16	15	eqcomi 2739	. . . . . 6 ⊢ 𝑣 = (Vtx‘⟨𝑣, 𝑒⟩)
17		eqid 2730	. . . . . 6 ⊢ (Edg‘⟨𝑣, 𝑒⟩) = (Edg‘⟨𝑣, 𝑒⟩)
18		eqid 2730	. . . . . 6 ⊢ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝} = {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝}
19		eqid 2730	. . . . . 6 ⊢ ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝})⟩ = ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝})⟩
20	16, 17, 18, 19	usgrres1 29249	. . . . 5 ⊢ ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝})⟩ ∈ USGraph)
21		eleq1 2817	. . . . . 6 ⊢ (𝑓 = ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝}) → (𝑓 ∈ Fin ↔ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝}) ∈ Fin))
22	21	adantl 481	. . . . 5 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝})) → (𝑓 ∈ Fin ↔ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝}) ∈ Fin))
23	13, 14	pm3.2i 470	. . . . . 6 ⊢ (𝑣 ∈ V ∧ 𝑒 ∈ V)
24		fusgrfisbase 29262	. . . . . 6 ⊢ (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = 0) → 𝑒 ∈ Fin)
25	23, 24	mp3an1 1450	. . . . 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) ∧ 𝑛 ∈ 𝑣))) → (𝑣 ∈ V ∧ 𝑒 ∈ V))
27		simprr1 1222	. . . . . . . . . 10 ⊢ (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))) → ⟨𝑣, 𝑒⟩ ∈ USGraph)
28		eleq1 2817	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑣) = (𝑦 + 1) → ((♯‘𝑣) ∈ ℕ0 ↔ (𝑦 + 1) ∈ ℕ0))
29		hashclb 14330	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ V → (𝑣 ∈ Fin ↔ (♯‘𝑣) ∈ ℕ0))
30	29	biimprd 248	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ V → ((♯‘𝑣) ∈ ℕ0 → 𝑣 ∈ Fin))
31	30	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ V ∧ 𝑒 ∈ V) → ((♯‘𝑣) ∈ ℕ0 → 𝑣 ∈ Fin))
32	31	com12 32	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑣) ∈ ℕ0 → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin))
33	28, 32	biimtrrdi 254	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑣) = (𝑦 + 1) → ((𝑦 + 1) ∈ ℕ0 → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin)))
34	33	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣) → ((𝑦 + 1) ∈ ℕ0 → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin)))
35	34	impcom 407	. . . . . . . . . . 11 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → ((𝑣 ∈ V ∧ 𝑒 ∈ V) → 𝑣 ∈ Fin))
36	35	impcom 407	. . . . . . . . . 10 ⊢ (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))) → 𝑣 ∈ Fin)
37		opfusgr 29257	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (⟨𝑣, 𝑒⟩ ∈ FinUSGraph ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ 𝑣 ∈ Fin)))
38	37	adantr 480	. . . . . . . . . 10 ⊢ (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))) → (⟨𝑣, 𝑒⟩ ∈ FinUSGraph ↔ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ 𝑣 ∈ Fin)))
39	27, 36, 38	mpbir2and 713	. . . . . . . . 9 ⊢ (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))) → ⟨𝑣, 𝑒⟩ ∈ FinUSGraph)
40		simprr3 1224	. . . . . . . . 9 ⊢ (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))) → 𝑛 ∈ 𝑣)
41	26, 39, 40	3jca 1128	. . . . . . . 8 ⊢ (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))) → ((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ FinUSGraph ∧ 𝑛 ∈ 𝑣))
42	23, 41	mpan 690	. . . . . . 7 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → ((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ FinUSGraph ∧ 𝑛 ∈ 𝑣))
43		fusgrfisstep 29263	. . . . . . 7 ⊢ (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ FinUSGraph ∧ 𝑛 ∈ 𝑣) → (( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝}) ∈ Fin → 𝑒 ∈ Fin))
44	42, 43	syl 17	. . . . . 6 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → (( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝}) ∈ Fin → 𝑒 ∈ Fin))
45	44	imp 406	. . . . 5 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝}) ∈ Fin) → 𝑒 ∈ Fin)
46	4, 8, 10, 12, 20, 22, 25, 45	opfi1ind 14484	. . . 4 ⊢ ((⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (iEdg‘𝐺) ∈ Fin)
47	3, 46	sylan 580	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (iEdg‘𝐺) ∈ Fin)
48		eqid 2730	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
49		eqid 2730	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
50	48, 49	usgredgffibi 29258	. . . 4 ⊢ (𝐺 ∈ USGraph → ((Edg‘𝐺) ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
51	50	adantr 480	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → ((Edg‘𝐺) ∈ Fin ↔ (iEdg‘𝐺) ∈ Fin))
52	47, 51	mpbird 257	. 2 ⊢ ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (Edg‘𝐺) ∈ Fin)
53	2, 52	sylbi 217	1 ⊢ (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∉ wnel 3030  {crab 3408  Vcvv 3450   ∖ cdif 3914  {csn 4592  ⟨cop 4598   ↦ cmpt 5191   I cid 5535   ↾ cres 5643  ‘cfv 6514  (class class class)co 7390  Fincfn 8921  0cc0 11075  1c1 11076   + caddc 11078  ℕ₀cn0 12449  ♯chash 14302  Vtxcvtx 28930  iEdgciedg 28931  Edgcedg 28981  USGraphcusgr 29083  FinUSGraphcfusgr 29250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-n0 12450  df-xnn0 12523  df-z 12537  df-uz 12801  df-fz 13476  df-hash 14303  df-vtx 28932  df-iedg 28933  df-edg 28982  df-uhgr 28992  df-upgr 29016  df-umgr 29017  df-uspgr 29084  df-usgr 29085  df-fusgr 29251

This theorem is referenced by:  fusgrfupgrfs  29265  nbfiusgrfi  29309  cusgrsizeindslem  29386  cusgrsizeinds  29387  sizusglecusglem2  29397  vtxdgfusgrf  29432  numclwwlk1  30297  clnbfiusgrfi  47848