Theorem cusgrsizeindslem 29520

Description: Lemma for cusgrsizeinds 29521. (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 29489	. . . . 5 ⊢ (𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)
2		cusgrsizeindb0.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	nbcplgr 29503	. . . . 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 6845	. 2 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑁)) = (♯‘(𝑉 ∖ {𝑁})))
7		cusgrusgr 29488	. . . . . 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 29440	. . . 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 29418	. . . . . . . 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 9181	. . . . . . 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 29387	. . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
22	20, 21	sylibr 234	. . . . . 6 ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
23		fusgrfis 29399	. . . . . . . 8 ⊢ (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin)
24	10, 23	eqeltrid 2841	. . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph → 𝐸 ∈ Fin)
25		rabfi 9181	. . . . . . 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 14311	. . . 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 14376	. . 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 3390   ∖ cdif 3887  {csn 4568  {cpr 4570  –1-1-onto→wf1o 6498  ‘cfv 6499  (class class class)co 7367  Fincfn 8893  1c1 11039   − cmin 11377  ♯chash 14292  Vtxcvtx 29065  Edgcedg 29116  USGraphcusgr 29218  FinUSGraphcfusgr 29385   NeighbVtx cnbgr 29401  ComplGraphccplgr 29478  ComplUSGraphccusgr 29479

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689  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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-fz 13462  df-hash 14293  df-vtx 29067  df-iedg 29068  df-edg 29117  df-uhgr 29127  df-upgr 29151  df-umgr 29152  df-uspgr 29219  df-usgr 29220  df-fusgr 29386  df-nbgr 29402  df-uvtx 29455  df-cplgr 29480  df-cusgr 29481

This theorem is referenced by:  cusgrsizeinds  29521