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Theorem cplgruvtxb 40637
Description: An graph is complete iff each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 1-Nov-2020.)
Hypothesis
Ref Expression
iscplgr.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
cplgruvtxb (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))

Proof of Theorem cplgruvtxb
Dummy variables 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscplgr.v . . 3 𝑉 = (Vtx‘𝐺)
21iscplgr 40636 . 2 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
31uvtxaisvtx 40615 . . . . . . . . 9 (𝑔 ∈ (UnivVtx‘𝐺) → 𝑔𝑉)
43adantl 481 . . . . . . . 8 ((𝐺𝑊𝑔 ∈ (UnivVtx‘𝐺)) → 𝑔𝑉)
54ralrimiva 2949 . . . . . . 7 (𝐺𝑊 → ∀𝑔 ∈ (UnivVtx‘𝐺)𝑔𝑉)
6 dfss3 3558 . . . . . . 7 ((UnivVtx‘𝐺) ⊆ 𝑉 ↔ ∀𝑔 ∈ (UnivVtx‘𝐺)𝑔𝑉)
75, 6sylibr 223 . . . . . 6 (𝐺𝑊 → (UnivVtx‘𝐺) ⊆ 𝑉)
87adantr 480 . . . . 5 ((𝐺𝑊 ∧ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)) → (UnivVtx‘𝐺) ⊆ 𝑉)
9 dfss3 3558 . . . . . . 7 (𝑉 ⊆ (UnivVtx‘𝐺) ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
109biimpri 217 . . . . . 6 (∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺) → 𝑉 ⊆ (UnivVtx‘𝐺))
1110adantl 481 . . . . 5 ((𝐺𝑊 ∧ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)) → 𝑉 ⊆ (UnivVtx‘𝐺))
128, 11eqssd 3585 . . . 4 ((𝐺𝑊 ∧ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)) → (UnivVtx‘𝐺) = 𝑉)
1312ex 449 . . 3 (𝐺𝑊 → (∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺) → (UnivVtx‘𝐺) = 𝑉))
14 raleleq 3133 . . . 4 (𝑉 = (UnivVtx‘𝐺) → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
1514eqcoms 2618 . . 3 ((UnivVtx‘𝐺) = 𝑉 → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
1613, 15impbid1 214 . 2 (𝐺𝑊 → (∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺) ↔ (UnivVtx‘𝐺) = 𝑉))
172, 16bitrd 267 1 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wss 3540  cfv 5804  Vtxcvtx 25673  UnivVtxcuvtxa 40551  ComplGraphccplgr 40552
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-sep 4709  ax-nul 4717  ax-pow 4769  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-ne 2782  df-ral 2901  df-rex 2902  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-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-uvtxa 40556  df-cplgr 40557
This theorem is referenced by:  cusgruvtxb  40644  nbcplgr  40656
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