Theorem fusgrfis 29415

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 2737	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	isfusgr 29403	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin))
3		usgrop 29248	. . . 4 ⊢ (𝐺 ∈ USGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph)
4		fvex 6855	. . . . 5 ⊢ (iEdg‘𝐺) ∈ V
5		mptresid 6018	. . . . . 6 ⊢ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝}) = (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝} ↦ 𝑞)
6		fvex 6855	. . . . . . 7 ⊢ (Edg‘⟨𝑣, 𝑒⟩) ∈ V
7	6	mptrabex 7181	. . . . . 6 ⊢ (𝑞 ∈ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝} ↦ 𝑞) ∈ V
8	5, 7	eqeltri 2833	. . . . 5 ⊢ ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝}) ∈ V
9		eleq1 2825	. . . . . 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 2825	. . . . . 6 ⊢ (𝑒 = 𝑓 → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
12	11	adantl 481	. . . . 5 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑒 ∈ Fin ↔ 𝑓 ∈ Fin))
13		vex 3446	. . . . . . . 8 ⊢ 𝑣 ∈ V
14		vex 3446	. . . . . . . 8 ⊢ 𝑒 ∈ V
15	13, 14	opvtxfvi 29094	. . . . . . 7 ⊢ (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣
16	15	eqcomi 2746	. . . . . 6 ⊢ 𝑣 = (Vtx‘⟨𝑣, 𝑒⟩)
17		eqid 2737	. . . . . 6 ⊢ (Edg‘⟨𝑣, 𝑒⟩) = (Edg‘⟨𝑣, 𝑒⟩)
18		eqid 2737	. . . . . 6 ⊢ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝} = {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝}
19		eqid 2737	. . . . . 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 29400	. . . . 5 ⊢ ((⟨𝑣, 𝑒⟩ ∈ USGraph ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑝 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑝})⟩ ∈ USGraph)
21		eleq1 2825	. . . . . 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 29413	. . . . . 6 ⊢ (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = 0) → 𝑒 ∈ Fin)
25	23, 24	mp3an1 1451	. . . . 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 1223	. . . . . . . . . 10 ⊢ (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))) → ⟨𝑣, 𝑒⟩ ∈ USGraph)
28		eleq1 2825	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑣) = (𝑦 + 1) → ((♯‘𝑣) ∈ ℕ0 ↔ (𝑦 + 1) ∈ ℕ0))
29		hashclb 14293	. . . . . . . . . . . . . . . . 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 1135	. . . . . . . . . . . 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 29408	. . . . . . . . . . 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 714	. . . . . . . . 9 ⊢ (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))) → ⟨𝑣, 𝑒⟩ ∈ FinUSGraph)
40		simprr3 1225	. . . . . . . . 9 ⊢ (((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))) → 𝑛 ∈ 𝑣)
41	26, 39, 40	3jca 1129	. . . . . . . 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 691	. . . . . . 7 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ USGraph ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → ((𝑣 ∈ V ∧ 𝑒 ∈ V) ∧ ⟨𝑣, 𝑒⟩ ∈ FinUSGraph ∧ 𝑛 ∈ 𝑣))
43		fusgrfisstep 29414	. . . . . . 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 14447	. . . 4 ⊢ ((⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (iEdg‘𝐺) ∈ Fin)
47	3, 46	sylan 581	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) ∈ Fin) → (iEdg‘𝐺) ∈ Fin)
48		eqid 2737	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
49		eqid 2737	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
50	48, 49	usgredgffibi 29409	. . . 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 1087   = wceq 1542   ∈ wcel 2114   ∉ wnel 3037  {crab 3401  Vcvv 3442   ∖ cdif 3900  {csn 4582  ⟨cop 4588   ↦ cmpt 5181   I cid 5526   ↾ cres 5634  ‘cfv 6500  (class class class)co 7368  Fincfn 8895  0cc0 11038  1c1 11039   + caddc 11041  ℕ₀cn0 12413  ♯chash 14265  Vtxcvtx 29081  iEdgciedg 29082  Edgcedg 29132  USGraphcusgr 29234  FinUSGraphcfusgr 29401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-n0 12414  df-xnn0 12487  df-z 12501  df-uz 12764  df-fz 13436  df-hash 14266  df-vtx 29083  df-iedg 29084  df-edg 29133  df-uhgr 29143  df-upgr 29167  df-umgr 29168  df-uspgr 29235  df-usgr 29236  df-fusgr 29402

This theorem is referenced by:  fusgrfupgrfs  29416  nbfiusgrfi  29460  cusgrsizeindslem  29537  cusgrsizeinds  29538  sizusglecusglem2  29548  vtxdgfusgrf  29583  numclwwlk1  30448  clnbfiusgrfi  48198