Theorem fusgrfis 29403

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 2736	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29391	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3		usgrop 29236	. . . 4 ⊢ (𝐺 ∈ USGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph)
4		fvex 6847	. . . . 5 ⊢ (iEdg‘𝐺) ∈ V
5		mptresid 6010	. . . . . 6 ⊢ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝}) = (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝} ↦ 𝑞)
6		fvex 6847	. . . . . . 7 ⊢ (Edg‘⟨𝑣, 𝑒⟩) ∈ V
7	6	mptrabex 7171	. . . . . 6 ⊢ (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝} ↦ 𝑞) ∈ V
8	5, 7	eqeltri 2832	. . . . 5 ⊢ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝}) ∈ V
9		eleq1 2824	. . . . . 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 2824	. . . . . 6 ⊢ (𝑒 = 𝑓 → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
12	11	adantl 481	. . . . 5 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
13		vex 3444	. . . . . . . 8 ⊢ 𝑣 ∈ V
14		vex 3444	. . . . . . . 8 ⊢ 𝑒 ∈ V
15	13, 14	opvtxfvi 29082	. . . . . . 7 ⊢ (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣
16	15	eqcomi 2745	. . . . . 6 ⊢ 𝑣 = (Vtx‘⟨𝑣, 𝑒⟩)
17		eqid 2736	. . . . . 6 ⊢ (Edg‘⟨𝑣, 𝑒⟩) = (Edg‘⟨𝑣, 𝑒⟩)
18		eqid 2736	. . . . . 6 ⊢ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝} = {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝}
19		eqid 2736	. . . . . 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 29388	. . . . 5 ⊢ ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝})⟩ ∈ USGraph)
21		eleq1 2824	. . . . . 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 29401	. . . . . 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 2824	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑣) = (𝑦 + 1) → ((♯‘𝑣) ∈ ℕ0 ↔ (𝑦 + 1) ∈ ℕ0))
29		hashclb 14281	. . . . . . . . . . . . . . . . 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 29396	. . . . . . . . . . 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 29402	. . . . . . 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 14435	. . . 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 2736	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
49		eqid 2736	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
50	48, 49	usgredgffibi 29397	. . . 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 1541   ∈ wcel 2113   ∉ wnel 3036  {crab 3399  Vcvv 3440   ∖ cdif 3898  {csn 4580  ⟨cop 4586   ↦ cmpt 5179   I cid 5518   ↾ cres 5626  ‘cfv 6492  (class class class)co 7358  Fincfn 8883  0cc0 11026  1c1 11027   + caddc 11029  ℕ₀cn0 12401  ♯chash 14253  Vtxcvtx 29069  iEdgciedg 29070  Edgcedg 29120  USGraphcusgr 29222  FinUSGraphcfusgr 29389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9813  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-n0 12402  df-xnn0 12475  df-z 12489  df-uz 12752  df-fz 13424  df-hash 14254  df-vtx 29071  df-iedg 29072  df-edg 29121  df-uhgr 29131  df-upgr 29155  df-umgr 29156  df-uspgr 29223  df-usgr 29224  df-fusgr 29390

This theorem is referenced by:  fusgrfupgrfs  29404  nbfiusgrfi  29448  cusgrsizeindslem  29525  cusgrsizeinds  29526  sizusglecusglem2  29536  vtxdgfusgrf  29571  numclwwlk1  30436  clnbfiusgrfi  48086