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Theorem iscplgr 29484

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 29482	. 2 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
3		eqss 3937	. . 3 ⊢ ((UnivVtx‘𝐺) = 𝑉 ↔ ((UnivVtx‘𝐺) ⊆ 𝑉 ∧ 𝑉 ⊆ (UnivVtx‘𝐺)))
4	1	uvtxssvtx 29459	. . . 4 ⊢ (UnivVtx‘𝐺) ⊆ 𝑉
5		dfss3 3910	. . . . 5 ⊢ (𝑉 ⊆ (UnivVtx‘𝐺) ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
6	5	anbi2i 624	. . . 4 ⊢ (((UnivVtx‘𝐺) ⊆ 𝑉 ∧ 𝑉 ⊆ (UnivVtx‘𝐺)) ↔ ((UnivVtx‘𝐺) ⊆ 𝑉 ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
7	4, 6	mpbiran 710	. . 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 1542 ∈ wcel 2114 ∀wral 3051 ⊆ wss 3889 ‘cfv 6498 Vtxcvtx 29065 UnivVtxcuvtx 29454 ComplGraphccplgr 29478

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pr 5375

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6454 df-fun 6500 df-fv 6506 df-ov 7370 df-uvtx 29455 df-cplgr 29480

This theorem is referenced by: iscplgrnb 29485 iscusgrvtx 29490 cplgr0 29494 cplgr0v 29496 cplgr1v 29499 cplgr2v 29501 cusgrexi 29512 structtocusgr 29515 cusgrres 29517

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