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Theorem cplgruvtxb 26192
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 26191 . 2 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
31uvtxaisvtx 26170 . . . . . . . . 9 (𝑔 ∈ (UnivVtx‘𝐺) → 𝑔𝑉)
43adantl 482 . . . . . . . 8 ((𝐺𝑊𝑔 ∈ (UnivVtx‘𝐺)) → 𝑔𝑉)
54ralrimiva 2965 . . . . . . 7 (𝐺𝑊 → ∀𝑔 ∈ (UnivVtx‘𝐺)𝑔𝑉)
6 dfss3 3578 . . . . . . 7 ((UnivVtx‘𝐺) ⊆ 𝑉 ↔ ∀𝑔 ∈ (UnivVtx‘𝐺)𝑔𝑉)
75, 6sylibr 224 . . . . . 6 (𝐺𝑊 → (UnivVtx‘𝐺) ⊆ 𝑉)
87adantr 481 . . . . 5 ((𝐺𝑊 ∧ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)) → (UnivVtx‘𝐺) ⊆ 𝑉)
9 dfss3 3578 . . . . . . 7 (𝑉 ⊆ (UnivVtx‘𝐺) ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
109biimpri 218 . . . . . 6 (∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺) → 𝑉 ⊆ (UnivVtx‘𝐺))
1110adantl 482 . . . . 5 ((𝐺𝑊 ∧ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)) → 𝑉 ⊆ (UnivVtx‘𝐺))
128, 11eqssd 3605 . . . 4 ((𝐺𝑊 ∧ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)) → (UnivVtx‘𝐺) = 𝑉)
1312ex 450 . . 3 (𝐺𝑊 → (∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺) → (UnivVtx‘𝐺) = 𝑉))
14 raleleq 3150 . . . 4 (𝑉 = (UnivVtx‘𝐺) → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
1514eqcoms 2634 . . 3 ((UnivVtx‘𝐺) = 𝑉 → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
1613, 15impbid1 215 . 2 (𝐺𝑊 → (∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺) ↔ (UnivVtx‘𝐺) = 𝑉))
172, 16bitrd 268 1 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wss 3560  cfv 5850  Vtxcvtx 25769  UnivVtxcuvtxa 26106  ComplGraphccplgr 26107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-uvtxa 26111  df-cplgr 26112
This theorem is referenced by:  cusgruvtxb  26199  nbcplgr  26211
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