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	⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 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.

Step	Hyp	Ref	Expression
1		fveq2 6103	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
2		iscplgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	1, 2	syl6eqr 2662	. . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
4		fveq2 6103	. . . 4 ⊢ (𝑔 = 𝐺 → (UnivVtx‘𝑔) = (UnivVtx‘𝐺))
5	4	eleq2d 2673	. . 3 ⊢ (𝑔 = 𝐺 → (𝑣 ∈ (UnivVtx‘𝑔) ↔ 𝑣 ∈ (UnivVtx‘𝐺)))
6	3, 5	raleqbidv 3129	. 2 ⊢ (𝑔 = 𝐺 → (∀𝑣 ∈ (Vtx‘𝑔)𝑣 ∈ (UnivVtx‘𝑔) ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
7		df-cplgr 40557	. 2 ⊢ ComplGraph = {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)𝑣 ∈ (UnivVtx‘𝑔)}
8	6, 7	elab2g 3322	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))

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