cplgruvtxb - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cplgruvtxb		Structured version Visualization version GIF version

Theorem cplgruvtxb 29349

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 6901	. . 3 ⊢ (𝑔 = 𝐺 → (UnivVtx‘𝑔) = (UnivVtx‘𝐺))
2		fveq2 6901	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
3		cplgruvtxb.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
4	2, 3	eqtr4di 2784	. . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
5	1, 4	eqeq12d 2742	. 2 ⊢ (𝑔 = 𝐺 → ((UnivVtx‘𝑔) = (Vtx‘𝑔) ↔ (UnivVtx‘𝐺) = 𝑉))
6		df-cplgr 29347	. 2 ⊢ ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)}
7	5, 6	elab2g 3668	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ‘cfv 6554 Vtxcvtx 28932 UnivVtxcuvtx 29321 ComplGraphccplgr 29345

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-iota 6506 df-fv 6562 df-cplgr 29347

This theorem is referenced by: iscplgr 29351 cusgruvtxb 29358 nbcplgr 29370

W3C validator

	
		OSZAR »