Theorem cusgrasizeindslem2 26003

Description: Lemma 2 for cusgrasizeinds 26004. (Contributed by Alexander van der Vekens, 11-Jan-2018.)

Hypothesis

Ref	Expression
cusgrares.f	⊢ 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑥)})

Assertion

Ref	Expression
cusgrasizeindslem2	⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) = ((#‘𝑉) − 1))

Distinct variable groups:   𝑥,𝐸   𝑥,𝑁   𝑥,𝑉

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem cusgrasizeindslem2

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

Step	Hyp	Ref	Expression
1		nbcusgra 25992	. . . 4 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁}))
2	1	3adant2 1073	. . 3 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁}))
3	2	fveq2d 6107	. 2 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = (#‘(𝑉 ∖ {𝑁})))
4		cusisusgra 25987	. . . . . 6 ⊢ (𝑉 ComplUSGrph 𝐸 → 𝑉 USGrph 𝐸)
5	4	anim1i 590	. . . . 5 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉))
6	5	3adant2 1073	. . . 4 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉))
7		nbgraf1o 25976	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑁)–1-1-onto→{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)})
8	6, 7	syl 17	. . 3 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑁)–1-1-onto→{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)})
9		nbusgra 25957	. . . . . . . 8 ⊢ (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
10	4, 9	syl 17	. . . . . . 7 ⊢ (𝑉 ComplUSGrph 𝐸 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
11	10	adantr 480	. . . . . 6 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
12		rabfi 8070	. . . . . . 7 ⊢ (𝑉 ∈ Fin → {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} ∈ Fin)
13	12	adantl 481	. . . . . 6 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin) → {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸} ∈ Fin)
14	11, 13	eqeltrd 2688	. . . . 5 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∈ Fin)
15	14	3adant3 1074	. . . 4 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∈ Fin)
16		usgrafis 25944	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) → 𝐸 ∈ Fin)
17	4, 16	sylan 487	. . . . . 6 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin) → 𝐸 ∈ Fin)
18		dmfi 8129	. . . . . . 7 ⊢ (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
19		rabfi 8070	. . . . . . 7 ⊢ (dom 𝐸 ∈ Fin → {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ∈ Fin)
20	18, 19	syl 17	. . . . . 6 ⊢ (𝐸 ∈ Fin → {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ∈ Fin)
21	17, 20	syl 17	. . . . 5 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin) → {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ∈ Fin)
22	21	3adant3 1074	. . . 4 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ∈ Fin)
23		hasheqf1o 12999	. . . 4 ⊢ (((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∈ Fin ∧ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ∈ Fin) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) ↔ ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑁)–1-1-onto→{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}))
24	15, 22, 23	syl2anc 691	. . 3 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) ↔ ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑁)–1-1-onto→{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}))
25	8, 24	mpbird 246	. 2 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}))
26		hashdifsn 13063	. . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (#‘(𝑉 ∖ {𝑁})) = ((#‘𝑉) − 1))
27	26	3adant1 1072	. 2 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (#‘(𝑉 ∖ {𝑁})) = ((#‘𝑉) − 1))
28	3, 25, 27	3eqtr3d 2652	1 ⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) = ((#‘𝑉) − 1))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ∉ wnel 2781  {crab 2900   ∖ cdif 3537  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583  dom cdm 5038  ran crn 5039   ↾ cres 5040  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  1c1 9816   − cmin 10145  #chash 12979   USGrph cusg 25859   Neighbors cnbgra 25946   ComplUSGrph ccusgra 25947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862  df-nbgra 25949  df-cusgra 25950

This theorem is referenced by:  cusgrasizeinds  26004