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.

Step	Hyp	Ref	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 ∧ ∀𝑥 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸)))
3	2	baibd 946	. . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → (𝑉 ComplUSGrph 𝐸 ↔ ∀𝑥 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸))
4		dfcleq 2604	. . . . 5 ⊢ ({𝑘 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} = 𝑉 ↔ ∀𝑥(𝑥 ∈ {𝑘 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ 𝑥 ∈ 𝑉))
5	4	a1i 11	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → ({𝑘 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} = 𝑉 ↔ ∀𝑥(𝑥 ∈ {𝑘 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ 𝑥 ∈ 𝑉)))
6		isuvtx 26016	. . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 UnivVertex 𝐸) = {𝑘 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸})
7	6	eqeq1d 2612	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑉 UnivVertex 𝐸) = 𝑉 ↔ {𝑘 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} = 𝑉))
8	7	adantr 480	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → ((𝑉 UnivVertex 𝐸) = 𝑉 ↔ {𝑘 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} = 𝑉))
9		df-ral 2901	. . . . 5 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸 ↔ ∀𝑥(𝑥 ∈ 𝑉 → ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸))
10		bicom 211	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ↔ (𝑥 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸)) ↔ ((𝑥 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸) ↔ 𝑥 ∈ 𝑉))
11	10	a1i 11	. . . . . . 7 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → ((𝑥 ∈ 𝑉 ↔ (𝑥 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸)) ↔ ((𝑥 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸) ↔ 𝑥 ∈ 𝑉)))
12		pm4.71 660	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 → ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸) ↔ (𝑥 ∈ 𝑉 ↔ (𝑥 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸)))
13	12	a1i 11	. . . . . . 7 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → ((𝑥 ∈ 𝑉 → ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸) ↔ (𝑥 ∈ 𝑉 ↔ (𝑥 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸))))
14		sneq 4135	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑥 → {𝑘} = {𝑥})
15	14	difeq2d 3690	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑥 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝑥}))
16		preq2 4213	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑥 → {𝑛, 𝑘} = {𝑛, 𝑥})
17	16	eleq1d 2672	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑥 → ({𝑛, 𝑘} ∈ ran 𝐸 ↔ {𝑛, 𝑥} ∈ ran 𝐸))
18	15, 17	raleqbidv 3129	. . . . . . . . . 10 ⊢ (𝑘 = 𝑥 → (∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸 ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸))
19	18	elrab 3331	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑘 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ (𝑥 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸))
20	19	a1i 11	. . . . . . . 8 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → (𝑥 ∈ {𝑘 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ (𝑥 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸)))
21	20	bibi1d 332	. . . . . . 7 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → ((𝑥 ∈ {𝑘 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ 𝑥 ∈ 𝑉) ↔ ((𝑥 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸) ↔ 𝑥 ∈ 𝑉)))
22	11, 13, 21	3bitr4d 299	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → ((𝑥 ∈ 𝑉 → ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸) ↔ (𝑥 ∈ {𝑘 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ 𝑥 ∈ 𝑉)))
23	22	albidv 1836	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → (∀𝑥(𝑥 ∈ 𝑉 → ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸) ↔ ∀𝑥(𝑥 ∈ {𝑘 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ 𝑥 ∈ 𝑉)))
24	9, 23	syl5bb 271	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → (∀𝑥 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸 ↔ ∀𝑥(𝑥 ∈ {𝑘 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸} ↔ 𝑥 ∈ 𝑉)))
25	5, 8, 24	3bitr4rd 300	. . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → (∀𝑥 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑥}){𝑛, 𝑥} ∈ ran 𝐸 ↔ (𝑉 UnivVertex 𝐸) = 𝑉))
26	3, 25	bitrd 267	. 2 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑉 USGrph 𝐸) → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 UnivVertex 𝐸) = 𝑉))
27	1, 26	mpancom 700	1 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 UnivVertex 𝐸) = 𝑉))

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