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Theorem cplgruvtxb 29445
Description: A graph 𝐺 is complete iff each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 15-Feb-2022.)
Hypothesis
Ref Expression
cplgruvtxb.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
cplgruvtxb (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))

Proof of Theorem cplgruvtxb
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . 3 (𝑔 = 𝐺 → (UnivVtx‘𝑔) = (UnivVtx‘𝐺))
2 fveq2 6907 . . . 4 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
3 cplgruvtxb.v . . . 4 𝑉 = (Vtx‘𝐺)
42, 3eqtr4di 2793 . . 3 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
51, 4eqeq12d 2751 . 2 (𝑔 = 𝐺 → ((UnivVtx‘𝑔) = (Vtx‘𝑔) ↔ (UnivVtx‘𝐺) = 𝑉))
6 df-cplgr 29443 . 2 ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)}
75, 6elab2g 3683 1 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  cfv 6563  Vtxcvtx 29028  UnivVtxcuvtx 29417  ComplGraphccplgr 29441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-cplgr 29443
This theorem is referenced by:  iscplgr  29447  cusgruvtxb  29454  nbcplgr  29466
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