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Theorem iscplgr 40631
Description: The property of being a complete graph. (Contributed by AV, 1-Nov-2020.)
Hypothesis
Ref Expression
iscplgr.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
iscplgr (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
Distinct variable groups:   𝑣,𝐺   𝑣,𝑉
Allowed substitution hint:   𝑊(𝑣)

Proof of Theorem iscplgr
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6088 . . . 4 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
2 iscplgr.v . . . 4 𝑉 = (Vtx‘𝐺)
31, 2syl6eqr 2661 . . 3 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
4 fveq2 6088 . . . 4 (𝑔 = 𝐺 → (UnivVtx‘𝑔) = (UnivVtx‘𝐺))
54eleq2d 2672 . . 3 (𝑔 = 𝐺 → (𝑣 ∈ (UnivVtx‘𝑔) ↔ 𝑣 ∈ (UnivVtx‘𝐺)))
63, 5raleqbidv 3128 . 2 (𝑔 = 𝐺 → (∀𝑣 ∈ (Vtx‘𝑔)𝑣 ∈ (UnivVtx‘𝑔) ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
7 df-cplgr 40552 . 2 ComplGraph = {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)𝑣 ∈ (UnivVtx‘𝑔)}
86, 7elab2g 3321 1 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wcel 1976  wral 2895  cfv 5790  Vtxcvtx 40224  UnivVtxcuvtxa 40546  ComplGraphccplgr 40547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-cplgr 40552
This theorem is referenced by:  cplgruvtxb  40632  iscplgrnb  40633  iscusgrvtx  40638  cplgr0  40642  cplgr0v  40644  cplgr1v  40647  cplgr2v  40649  cusgrexi  40657  cusgrres  40659
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