Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iscplgr Structured version   Visualization version   GIF version

Theorem iscplgr 40636
 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 6103 . . . 4 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
2 iscplgr.v . . . 4 𝑉 = (Vtx‘𝐺)
31, 2syl6eqr 2662 . . 3 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
4 fveq2 6103 . . . 4 (𝑔 = 𝐺 → (UnivVtx‘𝑔) = (UnivVtx‘𝐺))
54eleq2d 2673 . . 3 (𝑔 = 𝐺 → (𝑣 ∈ (UnivVtx‘𝑔) ↔ 𝑣 ∈ (UnivVtx‘𝐺)))
63, 5raleqbidv 3129 . 2 (𝑔 = 𝐺 → (∀𝑣 ∈ (Vtx‘𝑔)𝑣 ∈ (UnivVtx‘𝑔) ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
7 df-cplgr 40557 . 2 ComplGraph = {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)𝑣 ∈ (UnivVtx‘𝑔)}
86, 7elab2g 3322 1 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  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-iota 5768  df-fv 5812  df-cplgr 40557 This theorem is referenced by:  cplgruvtxb  40637  iscplgrnb  40638  iscusgrvtx  40643  cplgr0  40647  cplgr0v  40649  cplgr1v  40652  cplgr2v  40654  cusgrexi  40662  cusgrres  40664
 Copyright terms: Public domain W3C validator