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Theorem cusgrauvtxb 26024
 Description: An undirected simple graph is complete if and only if each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by Alexander van der Vekens, 18-Jan-2018.)
Assertion
Ref Expression
cusgrauvtxb (𝑉 USGrph 𝐸 → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 UnivVertex 𝐸) = 𝑉))

Proof of Theorem cusgrauvtxb
Dummy variables 𝑘 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrav 25867 . 2 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2 iscusgra 25985 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑥𝑉𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸)))
32baibd 946 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → (𝑉 ComplUSGrph 𝐸 ↔ ∀𝑥𝑉𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸))
4 dfcleq 2604 . . . . 5 ({𝑘𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} = 𝑉 ↔ ∀𝑥(𝑥 ∈ {𝑘𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ 𝑥𝑉))
54a1i 11 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → ({𝑘𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} = 𝑉 ↔ ∀𝑥(𝑥 ∈ {𝑘𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ 𝑥𝑉)))
6 isuvtx 26016 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 UnivVertex 𝐸) = {𝑘𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸})
76eqeq1d 2612 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑉 UnivVertex 𝐸) = 𝑉 ↔ {𝑘𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} = 𝑉))
87adantr 480 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → ((𝑉 UnivVertex 𝐸) = 𝑉 ↔ {𝑘𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} = 𝑉))
9 df-ral 2901 . . . . 5 (∀𝑥𝑉𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸 ↔ ∀𝑥(𝑥𝑉 → ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸))
10 bicom 211 . . . . . . . 8 ((𝑥𝑉 ↔ (𝑥𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸)) ↔ ((𝑥𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸) ↔ 𝑥𝑉))
1110a1i 11 . . . . . . 7 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → ((𝑥𝑉 ↔ (𝑥𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸)) ↔ ((𝑥𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸) ↔ 𝑥𝑉)))
12 pm4.71 660 . . . . . . . 8 ((𝑥𝑉 → ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸) ↔ (𝑥𝑉 ↔ (𝑥𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸)))
1312a1i 11 . . . . . . 7 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → ((𝑥𝑉 → ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸) ↔ (𝑥𝑉 ↔ (𝑥𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸))))
14 sneq 4135 . . . . . . . . . . . 12 (𝑘 = 𝑥 → {𝑘} = {𝑥})
1514difeq2d 3690 . . . . . . . . . . 11 (𝑘 = 𝑥 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝑥}))
16 preq2 4213 . . . . . . . . . . . 12 (𝑘 = 𝑥 → {𝑛, 𝑘} = {𝑛, 𝑥})
1716eleq1d 2672 . . . . . . . . . . 11 (𝑘 = 𝑥 → ({𝑛, 𝑘} ∈ ran 𝐸 ↔ {𝑛, 𝑥} ∈ ran 𝐸))
1815, 17raleqbidv 3129 . . . . . . . . . 10 (𝑘 = 𝑥 → (∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸))
1918elrab 3331 . . . . . . . . 9 (𝑥 ∈ {𝑘𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ (𝑥𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸))
2019a1i 11 . . . . . . . 8 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → (𝑥 ∈ {𝑘𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ (𝑥𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸)))
2120bibi1d 332 . . . . . . 7 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → ((𝑥 ∈ {𝑘𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ 𝑥𝑉) ↔ ((𝑥𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸) ↔ 𝑥𝑉)))
2211, 13, 213bitr4d 299 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → ((𝑥𝑉 → ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸) ↔ (𝑥 ∈ {𝑘𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ 𝑥𝑉)))
2322albidv 1836 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → (∀𝑥(𝑥𝑉 → ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸) ↔ ∀𝑥(𝑥 ∈ {𝑘𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ 𝑥𝑉)))
249, 23syl5bb 271 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → (∀𝑥𝑉𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸 ↔ ∀𝑥(𝑥 ∈ {𝑘𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ 𝑥𝑉)))
255, 8, 243bitr4rd 300 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → (∀𝑥𝑉𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸 ↔ (𝑉 UnivVertex 𝐸) = 𝑉))
263, 25bitrd 267 . 2 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 UnivVertex 𝐸) = 𝑉))
271, 26mpancom 700 1 (𝑉 USGrph 𝐸 → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 UnivVertex 𝐸) = 𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039  (class class class)co 6549   USGrph cusg 25859   ComplUSGrph ccusgra 25947   UnivVertex cuvtx 25948 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-usgra 25862  df-cusgra 25950  df-uvtx 25951 This theorem is referenced by:  vdiscusgra  26448
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