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Theorem iscplgr 30029

Description: The property of being a complete graph. (Contributed by AV, 1-Nov-2020.)

**Hypothesis**
Ref	Expression
cplgruvtxb.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**
Step	Hyp	Ref	Expression
1		cplgruvtxb.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	cplgruvtxb 30027	. 2 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
3		eqss 3928	. . 3 ⊢ ((UnivVtx‘𝐺) = 𝑉 ↔ ((UnivVtx‘𝐺) ⊆ 𝑉 ∧ 𝑉 ⊆ (UnivVtx‘𝐺)))
4	1	uvtxssvtx 30004	. . . 4 ⊢ (UnivVtx‘𝐺) ⊆ 𝑉
5		dfss3 3902	. . . . 5 ⊢ (𝑉 ⊆ (UnivVtx‘𝐺) ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
6	5	anbi2i 635	. . . 4 ⊢ (((UnivVtx‘𝐺) ⊆ 𝑉 ∧ 𝑉 ⊆ (UnivVtx‘𝐺)) ↔ ((UnivVtx‘𝐺) ⊆ 𝑉 ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
7	4, 6	mpbiran 722	. . 3 ⊢ (((UnivVtx‘𝐺) ⊆ 𝑉 ∧ 𝑉 ⊆ (UnivVtx‘𝐺)) ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
8	3, 7	bitri 278	. 2 ⊢ ((UnivVtx‘𝐺) = 𝑉 ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
9	2, 8	bitrdi 290	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ⊆ wss 3900 ‘cfv 6561 Vtxcvtx 29607 UnivVtxcuvtx 29999 ComplGraphccplgr 30023

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pr 5396

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-iota 6515 df-fun 6563 df-fv 6569 df-ov 7449 df-uvtx 30000 df-cplgr 30025

This theorem is used by: iscplgrnb 30030 iscusgrvtx 30035 cplgr0 30039 cplgr0v 30041 cplgr1v 30044 cplgr2v 30046 cusgrexi 30057 structtocusgr 30060 cusgrres 30062

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