Theorem cusgrsizeindslem 29530

Description: Lemma for cusgrsizeinds 29531. (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 29498	. . . . 5 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)
2		cusgrsizeindb0.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	nbcplgr 29512	. . . . 5 ⊢ ((𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
4	1, 3	sylan 581	. . . 4 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
5	4	3adant2 1132	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
6	5	fveq2d 6839	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘(𝑉 ∖ {𝑁})))
7		cusgrusgr 29497	. . . . . 6 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph)
8	7	anim1i 616	. . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉))
9	8	3adant2 1132	. . . 4 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉))
10		cusgrsizeindb0.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
11	2, 10	nbusgrf1o 29449	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒})
12	9, 11	syl 17	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒})
13	2, 10	nbusgr 29427	. . . . . . . 8 ⊢ (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
14	7, 13	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ ComplUSGraph → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
15	14	adantr 480	. . . . . 6 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸})
16		rabfi 9176	. . . . . . 7 ⊢ (𝑉 ∈ Fin → {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸} ∈ Fin)
17	16	adantl 481	. . . . . 6 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸} ∈ Fin)
18	15, 17	eqeltrd 2837	. . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 NeighbVtx 𝑁) ∈ Fin)
19	18	3adant3 1133	. . . 4 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) ∈ Fin)
20	7	anim1i 616	. . . . . . 7 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
21	2	isfusgr 29396	. . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
22	20, 21	sylibr 234	. . . . . 6 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
23		fusgrfis 29408	. . . . . . . 8 ⊢ (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
24	10, 23	eqeltrid 2841	. . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph → 𝐸 ∈ Fin)
25		rabfi 9176	. . . . . . 7 ⊢ (𝐸 ∈ Fin → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin)
26	24, 25	syl 17	. . . . . 6 ⊢ (𝐺 ∈ FinUSGraph → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin)
27	22, 26	syl 17	. . . . 5 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin)
28	27	3adant3 1133	. . . 4 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin)
29		hasheqf1o 14277	. . . 4 ⊢ (((𝐺 NeighbVtx 𝑁) ∈ Fin ∧ {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin) → ((♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ↔ ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}))
30	19, 28, 29	syl2anc 585	. . 3 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ↔ ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑁)–1-1-onto→{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}))
31	12, 30	mpbird 257	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}))
32		hashdifsn 14342	. . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − 1))
33	32	3adant1 1131	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − 1))
34	6, 31, 33	3eqtr3d 2780	1 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) = ((♯‘𝑉) − 1))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {crab 3400   ∖ cdif 3899  {csn 4581  {cpr 4583  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7361  Fincfn 8888  1c1 11032   − cmin 11369  ♯chash 14258  Vtxcvtx 29074  Edgcedg 29125  USGraphcusgr 29227  FinUSGraphcfusgr 29394   NeighbVtx cnbgr 29410  ComplGraphccplgr 29487  ComplUSGraphccusgr 29488

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 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683  ax-cnex 11087  ax-resscn 11088  ax-1cn 11089  ax-icn 11090  ax-addcl 11091  ax-addrcl 11092  ax-mulcl 11093  ax-mulrcl 11094  ax-mulcom 11095  ax-addass 11096  ax-mulass 11097  ax-distr 11098  ax-i2m1 11099  ax-1ne0 11100  ax-1rid 11101  ax-rnegex 11102  ax-rrecex 11103  ax-cnre 11104  ax-pre-lttri 11105  ax-pre-lttrn 11106  ax-pre-ltadd 11107  ax-pre-mulgt0 11108

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 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-oadd 8404  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-dju 9818  df-card 9856  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12151  df-2 12213  df-n0 12407  df-xnn0 12480  df-z 12494  df-uz 12757  df-fz 13429  df-hash 14259  df-vtx 29076  df-iedg 29077  df-edg 29126  df-uhgr 29136  df-upgr 29160  df-umgr 29161  df-uspgr 29228  df-usgr 29229  df-fusgr 29395  df-nbgr 29411  df-uvtx 29464  df-cplgr 29489  df-cusgr 29490

This theorem is referenced by:  cusgrsizeinds  29531