Theorem cusgrsizeindslem 30062

Description: Lemma for cusgrsizeinds 30063. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)

Hypotheses

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

Assertion

Ref	Expression
cusgrsizeindslem	⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) = ((♯‘𝑉) − 1))

Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑉

Proof of Theorem cusgrsizeindslem

Dummy variables 𝑓 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cusgrcplgr 30031	. . . . 5 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)
2		cusgrsizeindb0.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	nbcplgr 30045	. . . . 5 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
4	1, 3	sylan 592	. . . 4 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
5	4	3adant2 1149	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
6	5	fveq2d 6912	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘(𝑉 ∖ {𝑁})))
7		cusgrusgr 30030	. . . . . 6 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)
8	7	anim1i 627	. . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉))
9	8	3adant2 1149	. . . 4 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉))
10		cusgrsizeindb0.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
11	2, 10	nbusgrf1o 29982	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)⟶1-1onto{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒})
12	9, 11	syl 18	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)⟶1-1onto{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒})
13	2, 10	nbusgr 29960	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
14	7, 13	syl 18	. . . . . . 7 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
15	14	adantr 486	. . . . . 6 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
16		rabfi 9290	. . . . . . 7 ⊢ (𝑉 ∈ Fin → {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸} ∈ Fin)
17	16	adantl 487	. . . . . 6 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸} ∈ Fin)
18	15, 17	eqeltrd 2864	. . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 NeighbVtx 𝑁) ∈ Fin)
19	18	3adant3 1150	. . . 4 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) ∈ Fin)
20	7	anim1i 627	. . . . . . 7 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
21	2	isfusgr 29929	. . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
22	20, 21	sylibr 237	. . . . . 6 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
23		fusgrfis 29941	. . . . . . . 8 ⊢ (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
24	10, 23	eqeltrid 2868	. . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph → 𝐸 ∈ Fin)
25		rabfi 9290	. . . . . . 7 ⊢ (𝐸 ∈ Fin → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin)
26	24, 25	syl 18	. . . . . 6 ⊢ (𝐺 ∈ FinUSGraph → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin)
27	22, 26	syl 18	. . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin)
28	27	3adant3 1150	. . . 4 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin)
29		hasheqf1o 14520	. . . 4 ⊢ (((𝐺 NeighbVtx 𝑁) ∈ Fin ∧ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin) → ((♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ↔ ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)⟶1-1onto{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}))
30	19, 28, 29	syl2anc 596	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ↔ ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)⟶1-1onto{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}))
31	12, 30	mpbird 260	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}))
32		hashdifsn 14586	. . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − 1))
33	32	3adant1 1148	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − 1))
34	6, 31, 33	3eqtr3d 2806	1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) = ((♯‘𝑉) − 1))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {crab 3416   ∖ cdif 4048  {csn 4588  {cpr 4590  ⟶1-1ontowf1o 6557  ‘cfv 6558  (class class class)co 7443  Fincfn 9001  1c1 11228   − cmin 11568  ♯chash 14501  Vtxcvtx 29604  Edgcedg 29655  USGraphcusgr 29760  FinUSGraphcfusgr 29927   NeighbVtx cnbgr 29943  ComplGraphccplgr 30020  ComplUSGraphccusgr 30021

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-dju 10009  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-n0 12634  df-xnn0 12707  df-z 12721  df-uz 12993  df-fz 13667  df-hash 14502  df-vtx 29606  df-iedg 29607  df-edg 29656  df-uhgr 29666  df-upgr 29690  df-umgr 29691  df-uspgr 29761  df-usgr 29762  df-fusgr 29928  df-nbgr 29944  df-uvtx 29997  df-cplgr 30022  df-cusgr 30023

This theorem is used by:  cusgrsizeinds  30063