Theorem cplgruvtxb 29614

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 6867	. . 3 ⊢ (𝑔 = 𝐺 → (UnivVtx‘𝑔) = (UnivVtx‘𝐺))
2		fveq2 6867	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
3		cplgruvtxb.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
4	2, 3	eqtr4di 2815	. . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
5	1, 4	eqeq12d 2778	. 2 ⊢ (𝑔 = 𝐺 → ((UnivVtx‘𝑔) = (Vtx‘𝑔) ↔ (UnivVtx‘𝐺) = 𝑉))
6		df-cplgr 29612	. 2 ⊢ ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)}
7	5, 6	elab2g 3639	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1560   ∈ wcel 2142  ‘cfv 6521  Vtxcvtx 29197  UnivVtxcuvtx 29586  ComplGraphccplgr 29610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6477  df-fv 6529  df-cplgr 29612

This theorem is referenced by:  iscplgr  29616  cusgruvtxb  29623  nbcplgr  29635