Theorem cplgruvtxb 29347

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	⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))

Proof of Theorem cplgruvtxb

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6861	. . 3 ⊢ (𝑔 = 𝐺 → (UnivVtx‘𝑔) = (UnivVtx‘𝐺))
2		fveq2 6861	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
3		cplgruvtxb.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
4	2, 3	eqtr4di 2783	. . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
5	1, 4	eqeq12d 2746	. 2 ⊢ (𝑔 = 𝐺 → ((UnivVtx‘𝑔) = (Vtx‘𝑔) ↔ (UnivVtx‘𝐺) = 𝑉))
6		df-cplgr 29345	. 2 ⊢ ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)}
7	5, 6	elab2g 3650	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ‘cfv 6514  Vtxcvtx 28930  UnivVtxcuvtx 29319  ComplGraphccplgr 29343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-cplgr 29345

This theorem is referenced by:  iscplgr  29349  cusgruvtxb  29356  nbcplgr  29368