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Theorem cplgruvtxb 26542
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 6332 . . 3 (𝑔 = 𝐺 → (UnivVtx‘𝑔) = (UnivVtx‘𝐺))
2 fveq2 6332 . . . 4 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
3 cplgruvtxb.v . . . 4 𝑉 = (Vtx‘𝐺)
42, 3syl6eqr 2823 . . 3 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
51, 4eqeq12d 2786 . 2 (𝑔 = 𝐺 → ((UnivVtx‘𝑔) = (Vtx‘𝑔) ↔ (UnivVtx‘𝐺) = 𝑉))
6 df-cplgr 26540 . 2 ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)}
75, 6elab2g 3504 1 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  cfv 6031  Vtxcvtx 26094  UnivVtxcuvtx 26509  ComplGraphccplgr 26538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-cplgr 26540
This theorem is referenced by:  iscplgr  26544  cusgruvtxb  26552  nbcplgr  26564
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