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Theorem iscplgr 29883

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 29881	. 2 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
3		eqss 3949	. . 3 ⊢ ((UnivVtx‘𝐺) = 𝑉 ↔ ((UnivVtx‘𝐺) ⊆ 𝑉 ∧ 𝑉 ⊆ (UnivVtx‘𝐺)))
4	1	uvtxssvtx 29858	. . . 4 ⊢ (UnivVtx‘𝐺) ⊆ 𝑉
5		dfss3 3923	. . . . 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 2145 ∀wral 3078 ⊆ wss 3902 ‘cfv 6537 Vtxcvtx 29461 UnivVtxcuvtx 29853 ComplGraphccplgr 29877

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pr 5402

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 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-iota 6493 df-fun 6539 df-fv 6545 df-ov 7420 df-uvtx 29854 df-cplgr 29879

This theorem is used by: iscplgrnb 29884 iscusgrvtx 29889 cplgr0 29893 cplgr0v 29895 cplgr1v 29898 cplgr2v 29900 cusgrexi 29911 structtocusgr 29914 cusgrres 29916

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