Theorem cusgrsize 26350

Description: The size of a finite complete simple graph with 𝑛 vertices (𝑛 ∈ ℕ₀) is (𝑛C2) ("𝑛 choose 2") resp. (((𝑛 − 1)∗𝑛) / 2), see definition in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 10-Nov-2020.)

Hypotheses

Ref	Expression
cusgrsizeindb0.v	⊢ 𝑉 = (Vtx‘𝐺)
cusgrsizeindb0.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
cusgrsize	⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘𝐸) = ((#‘𝑉)C2))

Proof of Theorem cusgrsize

Dummy variables 𝑒 𝑓 𝑛 𝑣 𝑐 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cusgrsizeindb0.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
2		edgval 25941	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3	1, 2	eqtri 2644	. . . 4 ⊢ 𝐸 = ran (iEdg‘𝐺)
4	3	a1i 11	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐸 = ran (iEdg‘𝐺))
5	4	fveq2d 6195	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘𝐸) = (#‘ran (iEdg‘𝐺)))
6		cusgrsizeindb0.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
7	6	opeq1i 4405	. . . 4 ⊢ ⟨𝑉, (iEdg‘𝐺)⟩ = ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩
8		cusgrop 26334	. . . 4 ⊢ (𝐺 ∈ ComplUSGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplUSGraph)
9	7, 8	syl5eqel 2705	. . 3 ⊢ (𝐺 ∈ ComplUSGraph → ⟨𝑉, (iEdg‘𝐺)⟩ ∈ ComplUSGraph)
10		fvex 6201	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
11		fvex 6201	. . . . 5 ⊢ (Edg‘⟨𝑣, 𝑒⟩) ∈ V
12		rabexg 4812	. . . . . 6 ⊢ ((Edg‘⟨𝑣, 𝑒⟩) ∈ V → {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐} ∈ V)
13	12	resiexd 6480	. . . . 5 ⊢ ((Edg‘⟨𝑣, 𝑒⟩) ∈ V → ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) ∈ V)
14	11, 13	ax-mp 5	. . . 4 ⊢ ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) ∈ V
15		rneq 5351	. . . . . 6 ⊢ (𝑒 = (iEdg‘𝐺) → ran 𝑒 = ran (iEdg‘𝐺))
16	15	fveq2d 6195	. . . . 5 ⊢ (𝑒 = (iEdg‘𝐺) → (#‘ran 𝑒) = (#‘ran (iEdg‘𝐺)))
17		fveq2 6191	. . . . . 6 ⊢ (𝑣 = 𝑉 → (#‘𝑣) = (#‘𝑉))
18	17	oveq1d 6665	. . . . 5 ⊢ (𝑣 = 𝑉 → ((#‘𝑣)C2) = ((#‘𝑉)C2))
19	16, 18	eqeqan12rd 2640	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = (iEdg‘𝐺)) → ((#‘ran 𝑒) = ((#‘𝑣)C2) ↔ (#‘ran (iEdg‘𝐺)) = ((#‘𝑉)C2)))
20		rneq 5351	. . . . . 6 ⊢ (𝑒 = 𝑓 → ran 𝑒 = ran 𝑓)
21	20	fveq2d 6195	. . . . 5 ⊢ (𝑒 = 𝑓 → (#‘ran 𝑒) = (#‘ran 𝑓))
22		fveq2 6191	. . . . . 6 ⊢ (𝑣 = 𝑤 → (#‘𝑣) = (#‘𝑤))
23	22	oveq1d 6665	. . . . 5 ⊢ (𝑣 = 𝑤 → ((#‘𝑣)C2) = ((#‘𝑤)C2))
24	21, 23	eqeqan12rd 2640	. . . 4 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((#‘ran 𝑒) = ((#‘𝑣)C2) ↔ (#‘ran 𝑓) = ((#‘𝑤)C2)))
25		vex 3203	. . . . . . 7 ⊢ 𝑣 ∈ V
26		vex 3203	. . . . . . 7 ⊢ 𝑒 ∈ V
27	25, 26	opvtxfvi 25889	. . . . . 6 ⊢ (Vtx‘⟨𝑣, 𝑒⟩) = 𝑣
28	27	eqcomi 2631	. . . . 5 ⊢ 𝑣 = (Vtx‘⟨𝑣, 𝑒⟩)
29		eqid 2622	. . . . 5 ⊢ (Edg‘⟨𝑣, 𝑒⟩) = (Edg‘⟨𝑣, 𝑒⟩)
30		eqid 2622	. . . . 5 ⊢ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐} = {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}
31		eqid 2622	. . . . 5 ⊢ ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})⟩ = ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})⟩
32	28, 29, 30, 31	cusgrres 26344	. . . 4 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})⟩ ∈ ComplUSGraph)
33		rneq 5351	. . . . . . 7 ⊢ (𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) → ran 𝑓 = ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}))
34	33	fveq2d 6195	. . . . . 6 ⊢ (𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) → (#‘ran 𝑓) = (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})))
35	34	adantl 482	. . . . 5 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → (#‘ran 𝑓) = (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})))
36		fveq2 6191	. . . . . . 7 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
37	36	adantr 481	. . . . . 6 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
38	37	oveq1d 6665	. . . . 5 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → ((#‘𝑤)C2) = ((#‘(𝑣 ∖ {𝑛}))C2))
39	35, 38	eqeq12d 2637	. . . 4 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) → ((#‘ran 𝑓) = ((#‘𝑤)C2) ↔ (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) = ((#‘(𝑣 ∖ {𝑛}))C2)))
40		edgopval 25944	. . . . . . . . 9 ⊢ ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
41	25, 26, 40	mp2an 708	. . . . . . . 8 ⊢ (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒
42	41	a1i 11	. . . . . . 7 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = 0) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
43	42	eqcomd 2628	. . . . . 6 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = 0) → ran 𝑒 = (Edg‘⟨𝑣, 𝑒⟩))
44	43	fveq2d 6195	. . . . 5 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = 0) → (#‘ran 𝑒) = (#‘(Edg‘⟨𝑣, 𝑒⟩)))
45		cusgrusgr 26315	. . . . . . 7 ⊢ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph → ⟨𝑣, 𝑒⟩ ∈ USGraph )
46		usgruhgr 26078	. . . . . . 7 ⊢ (⟨𝑣, 𝑒⟩ ∈ USGraph → ⟨𝑣, 𝑒⟩ ∈ UHGraph )
47	45, 46	syl 17	. . . . . 6 ⊢ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph → ⟨𝑣, 𝑒⟩ ∈ UHGraph )
48	28, 29	cusgrsizeindb0 26345	. . . . . 6 ⊢ ((⟨𝑣, 𝑒⟩ ∈ UHGraph ∧ (#‘𝑣) = 0) → (#‘(Edg‘⟨𝑣, 𝑒⟩)) = ((#‘𝑣)C2))
49	47, 48	sylan 488	. . . . 5 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = 0) → (#‘(Edg‘⟨𝑣, 𝑒⟩)) = ((#‘𝑣)C2))
50	44, 49	eqtrd 2656	. . . 4 ⊢ ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = 0) → (#‘ran 𝑒) = ((#‘𝑣)C2))
51		rnresi 5479	. . . . . . . . . 10 ⊢ ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) = {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}
52	51	fveq2i 6194	. . . . . . . . 9 ⊢ (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) = (#‘{𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})
53	41	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → (Edg‘⟨𝑣, 𝑒⟩) = ran 𝑒)
54	53	rabeqdv 3194	. . . . . . . . . 10 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐} = {𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐})
55	54	fveq2d 6195	. . . . . . . . 9 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → (#‘{𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐}) = (#‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}))
56	52, 55	syl5eq 2668	. . . . . . . 8 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) = (#‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}))
57	56	eqeq1d 2624	. . . . . . 7 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → ((#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) = ((#‘(𝑣 ∖ {𝑛}))C2) ↔ (#‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}) = ((#‘(𝑣 ∖ {𝑛}))C2)))
58	57	biimpd 219	. . . . . 6 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) → ((#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) = ((#‘(𝑣 ∖ {𝑛}))C2) → (#‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}) = ((#‘(𝑣 ∖ {𝑛}))C2)))
59	58	imdistani 726	. . . . 5 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) = ((#‘(𝑣 ∖ {𝑛}))C2)) → (((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ (#‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}) = ((#‘(𝑣 ∖ {𝑛}))C2)))
60	41	eqcomi 2631	. . . . . . 7 ⊢ ran 𝑒 = (Edg‘⟨𝑣, 𝑒⟩)
61		eqid 2622	. . . . . . 7 ⊢ {𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐} = {𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}
62	28, 60, 61	cusgrsize2inds 26349	. . . . . 6 ⊢ ((𝑦 + 1) ∈ ℕ0 → ((⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣) → ((#‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}) = ((#‘(𝑣 ∖ {𝑛}))C2) → (#‘ran 𝑒) = ((#‘𝑣)C2))))
63	62	imp31 448	. . . . 5 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ (#‘{𝑐 ∈ ran 𝑒 ∣ 𝑛 ∉ 𝑐}) = ((#‘(𝑣 ∖ {𝑛}))C2)) → (#‘ran 𝑒) = ((#‘𝑣)C2))
64	59, 63	syl 17	. . . 4 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ ComplUSGraph ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ (#‘ran ( I ↾ {𝑐 ∈ (Edg‘⟨𝑣, 𝑒⟩) ∣ 𝑛 ∉ 𝑐})) = ((#‘(𝑣 ∖ {𝑛}))C2)) → (#‘ran 𝑒) = ((#‘𝑣)C2))
65	10, 14, 19, 24, 32, 39, 50, 64	opfi1ind 13284	. . 3 ⊢ ((⟨𝑉, (iEdg‘𝐺)⟩ ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘ran (iEdg‘𝐺)) = ((#‘𝑉)C2))
66	9, 65	sylan 488	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘ran (iEdg‘𝐺)) = ((#‘𝑉)C2))
67	5, 66	eqtrd 2656	1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘𝐸) = ((#‘𝑉)C2))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ∉ wnel 2897  {crab 2916  Vcvv 3200   ∖ cdif 3571  {csn 4177  ⟨cop 4183   I cid 5023  ran crn 5115   ↾ cres 5116  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  0cc0 9936  1c1 9937   + caddc 9939  2c2 11070  ℕ₀cn0 11292  Ccbc 13089  #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   USGraph cusgr 26044  ComplUSGraphccusgr 26227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-fac 13061  df-bc 13090  df-hash 13118  df-vtx 25876  df-iedg 25877  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-fusgr 26209  df-nbgr 26228  df-uvtxa 26230  df-cplgr 26231  df-cusgr 26232

This theorem is referenced by:  fusgrmaxsize  26360