Theorem iscplgr 26310

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 6191	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
2		iscplgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	1, 2	syl6eqr 2674	. . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
4		fveq2 6191	. . . 4 ⊢ (𝑔 = 𝐺 → (UnivVtx‘𝑔) = (UnivVtx‘𝐺))
5	4	eleq2d 2687	. . 3 ⊢ (𝑔 = 𝐺 → (𝑣 ∈ (UnivVtx‘𝑔) ↔ 𝑣 ∈ (UnivVtx‘𝐺)))
6	3, 5	raleqbidv 3152	. 2 ⊢ (𝑔 = 𝐺 → (∀𝑣 ∈ (Vtx‘𝑔)𝑣 ∈ (UnivVtx‘𝑔) ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
7		df-cplgr 26231	. 2 ⊢ ComplGraph = {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)𝑣 ∈ (UnivVtx‘𝑔)}
8	6, 7	elab2g 3353	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ‘cfv 5888  Vtxcvtx 25874  UnivVtxcuvtxa 26225  ComplGraphccplgr 26226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-cplgr 26231

This theorem is referenced by:  cplgruvtxb  26311  iscplgrnb  26312  iscusgrvtx  26317  cplgr0  26321  cplgr0v  26323  cplgr1v  26326  cplgr2v  26328  cusgrexi  26339  structtocusgr  26342  cusgrres  26344