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Theorem iscplgr 29432

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 29430	. 2 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
3		eqss 3999	. . 3 ⊢ ((UnivVtx‘𝐺) = 𝑉 ↔ ((UnivVtx‘𝐺) ⊆ 𝑉 ∧ 𝑉 ⊆ (UnivVtx‘𝐺)))
4	1	uvtxssvtx 29407	. . . 4 ⊢ (UnivVtx‘𝐺) ⊆ 𝑉
5		dfss3 3972	. . . . 5 ⊢ (𝑉 ⊆ (UnivVtx‘𝐺) ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
6	5	anbi2i 623	. . . 4 ⊢ (((UnivVtx‘𝐺) ⊆ 𝑉 ∧ 𝑉 ⊆ (UnivVtx‘𝐺)) ↔ ((UnivVtx‘𝐺) ⊆ 𝑉 ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
7	4, 6	mpbiran 709	. . 3 ⊢ (((UnivVtx‘𝐺) ⊆ 𝑉 ∧ 𝑉 ⊆ (UnivVtx‘𝐺)) ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
8	3, 7	bitri 275	. 2 ⊢ ((UnivVtx‘𝐺) = 𝑉 ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
9	2, 8	bitrdi 287	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ‘cfv 6561 Vtxcvtx 29013 UnivVtxcuvtx 29402 ComplGraphccplgr 29426

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-uvtx 29403 df-cplgr 29428

This theorem is referenced by: iscplgrnb 29433 iscusgrvtx 29438 cplgr0 29442 cplgr0v 29444 cplgr1v 29447 cplgr2v 29449 cusgrexi 29460 structtocusgr 29463 cusgrres 29466

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