Theorem isgrtri 48256

Description: A triangle in a graph. (Contributed by AV, 20-Jul-2025.)

Hypotheses

Ref	Expression
grtri.v	⊢ 𝑉 = (Vtx‘𝐺)
grtri.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
isgrtri	⊢ (𝑇 ∈ (GrTriangles‘𝐺) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))

Distinct variable groups:   𝑥,𝐸,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧   𝑥,𝑉,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧

Proof of Theorem isgrtri

Dummy variables 𝑓 𝑡 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grtri.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		grtri.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	grtriprop 48254	. 2 ⊢ (𝑇 ∈ (GrTriangles‘𝐺) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
4		df-tp 4586	. . . . . . . . . 10 ⊢ {𝑥, 𝑦, 𝑧} = ({𝑥, 𝑦} ∪ {𝑧})
5		prelpwi 5396	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → {𝑥, 𝑦} ∈ 𝒫 𝑉)
6		snelpwi 5393	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑉 → {𝑧} ∈ 𝒫 𝑉)
7	5, 6	anim12i 614	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑉) → ({𝑥, 𝑦} ∈ 𝒫 𝑉 ∧ {𝑧} ∈ 𝒫 𝑉))
8	7	anasss 466	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ({𝑥, 𝑦} ∈ 𝒫 𝑉 ∧ {𝑧} ∈ 𝒫 𝑉))
9		pwuncl 7717	. . . . . . . . . . 11 ⊢ (({𝑥, 𝑦} ∈ 𝒫 𝑉 ∧ {𝑧} ∈ 𝒫 𝑉) → ({𝑥, 𝑦} ∪ {𝑧}) ∈ 𝒫 𝑉)
10	8, 9	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ({𝑥, 𝑦} ∪ {𝑧}) ∈ 𝒫 𝑉)
11	4, 10	eqeltrid 2841	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → {𝑥, 𝑦, 𝑧} ∈ 𝒫 𝑉)
12	11	adantr 480	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))) → {𝑥, 𝑦, 𝑧} ∈ 𝒫 𝑉)
13		eleq1 2825	. . . . . . . . . 10 ⊢ (𝑇 = {𝑥, 𝑦, 𝑧} → (𝑇 ∈ 𝒫 𝑉 ↔ {𝑥, 𝑦, 𝑧} ∈ 𝒫 𝑉))
14	13	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → (𝑇 ∈ 𝒫 𝑉 ↔ {𝑥, 𝑦, 𝑧} ∈ 𝒫 𝑉))
15	14	adantl 481	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))) → (𝑇 ∈ 𝒫 𝑉 ↔ {𝑥, 𝑦, 𝑧} ∈ 𝒫 𝑉))
16	12, 15	mpbird 257	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))) → 𝑇 ∈ 𝒫 𝑉)
17		ovex 7393	. . . . . . . . . 10 ⊢ (0..^3) ∈ V
18	17	mptex 7171	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))) ∈ V
19	18	a1i 11	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))) → (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))) ∈ V)
20		3anass 1095	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) ↔ (𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)))
21	20	biimpri 228	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉))
22		fveq2 6835	. . . . . . . . . . . . . 14 ⊢ (𝑇 = {𝑥, 𝑦, 𝑧} → (♯‘𝑇) = (♯‘{𝑥, 𝑦, 𝑧}))
23	22	eqcomd 2743	. . . . . . . . . . . . 13 ⊢ (𝑇 = {𝑥, 𝑦, 𝑧} → (♯‘{𝑥, 𝑦, 𝑧}) = (♯‘𝑇))
24	23	3ad2ant1 1134	. . . . . . . . . . . 12 ⊢ ((𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → (♯‘{𝑥, 𝑦, 𝑧}) = (♯‘𝑇))
25		simp2 1138	. . . . . . . . . . . 12 ⊢ ((𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → (♯‘𝑇) = 3)
26	24, 25	eqtrd 2772	. . . . . . . . . . 11 ⊢ ((𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → (♯‘{𝑥, 𝑦, 𝑧}) = 3)
27		eqid 2737	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))) = (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))
28		eqid 2737	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑦, 𝑧} = {𝑥, 𝑦, 𝑧}
29	27, 28	tpf1o 14428	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) ∧ (♯‘{𝑥, 𝑦, 𝑧}) = 3) → (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))):(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧})
30	21, 26, 29	syl2an 597	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))) → (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))):(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧})
31		f1oeq3 6765	. . . . . . . . . . . 12 ⊢ (𝑇 = {𝑥, 𝑦, 𝑧} → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))):(0..^3)–1-1-onto→𝑇 ↔ (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))):(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧}))
32	31	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))):(0..^3)–1-1-onto→𝑇 ↔ (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))):(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧}))
33	32	adantl 481	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))) → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))):(0..^3)–1-1-onto→𝑇 ↔ (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))):(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧}))
34	30, 33	mpbird 257	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))) → (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))):(0..^3)–1-1-onto→𝑇)
35	27	tpf1ofv0 14423	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑉 → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0) = 𝑥)
36	35	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0) = 𝑥)
37	27	tpf1ofv1 14424	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ 𝑉 → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1) = 𝑦)
38	37	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1) = 𝑦)
39	38	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1) = 𝑦)
40	36, 39	preq12d 4699	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1)} = {𝑥, 𝑦})
41	40	eqcomd 2743	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → {𝑥, 𝑦} = {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1)})
42	41	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ({𝑥, 𝑦} ∈ 𝐸 ↔ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1)} ∈ 𝐸))
43	27	tpf1ofv2 14425	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝑉 → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2) = 𝑧)
44	43	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2) = 𝑧)
45	44	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2) = 𝑧)
46	36, 45	preq12d 4699	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} = {𝑥, 𝑧})
47	46	eqcomd 2743	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → {𝑥, 𝑧} = {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)})
48	47	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ({𝑥, 𝑧} ∈ 𝐸 ↔ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸))
49	39, 45	preq12d 4699	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} = {𝑦, 𝑧})
50	49	eqcomd 2743	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → {𝑦, 𝑧} = {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)})
51	50	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ({𝑦, 𝑧} ∈ 𝐸 ↔ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸))
52	42, 48, 51	3anbi123d 1439	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) ↔ ({((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1)} ∈ 𝐸 ∧ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸 ∧ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸)))
53	52	biimpd 229	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → ({((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1)} ∈ 𝐸 ∧ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸 ∧ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸)))
54	53	2a1d 26	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑇 = {𝑥, 𝑦, 𝑧} → ((♯‘𝑇) = 3 → (({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸) → ({((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1)} ∈ 𝐸 ∧ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸 ∧ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸)))))
55	54	3imp2 1351	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))) → ({((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1)} ∈ 𝐸 ∧ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸 ∧ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸))
56	34, 55	jca 511	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))) → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))):(0..^3)–1-1-onto→𝑇 ∧ ({((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1)} ∈ 𝐸 ∧ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸 ∧ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸)))
57		f1oeq1 6763	. . . . . . . . 9 ⊢ (𝑓 = (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))) → (𝑓:(0..^3)–1-1-onto→𝑇 ↔ (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))):(0..^3)–1-1-onto→𝑇))
58		fveq1 6834	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))) → (𝑓‘0) = ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0))
59		fveq1 6834	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))) → (𝑓‘1) = ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1))
60	58, 59	preq12d 4699	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))) → {(𝑓‘0), (𝑓‘1)} = {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1)})
61	60	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑓 = (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))) → ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ↔ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1)} ∈ 𝐸))
62		fveq1 6834	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))) → (𝑓‘2) = ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2))
63	58, 62	preq12d 4699	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))) → {(𝑓‘0), (𝑓‘2)} = {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)})
64	63	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑓 = (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))) → ({(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ↔ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸))
65	59, 62	preq12d 4699	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))) → {(𝑓‘1), (𝑓‘2)} = {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)})
66	65	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑓 = (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))) → ({(𝑓‘1), (𝑓‘2)} ∈ 𝐸 ↔ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸))
67	61, 64, 66	3anbi123d 1439	. . . . . . . . 9 ⊢ (𝑓 = (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))) → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) ↔ ({((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1)} ∈ 𝐸 ∧ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸 ∧ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸)))
68	57, 67	anbi12d 633	. . . . . . . 8 ⊢ (𝑓 = (𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))) → ((𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ↔ ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧))):(0..^3)–1-1-onto→𝑇 ∧ ({((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1)} ∈ 𝐸 ∧ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸 ∧ {((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1), ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2)} ∈ 𝐸))))
69	19, 56, 68	spcedv 3553	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))) → ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)))
70	16, 69	jca 511	. . . . . 6 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))) → (𝑇 ∈ 𝒫 𝑉 ∧ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))))
71	1	1vgrex 29079	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑉 → 𝐺 ∈ V)
72	1, 2	grtri 48253	. . . . . . . . . . 11 ⊢ (𝐺 ∈ V → (GrTriangles‘𝐺) = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))})
73	72	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝐺 ∈ V → (𝑇 ∈ (GrTriangles‘𝐺) ↔ 𝑇 ∈ {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))}))
74	71, 73	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑉 → (𝑇 ∈ (GrTriangles‘𝐺) ↔ 𝑇 ∈ {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))}))
75		f1oeq3 6765	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (𝑓:(0..^3)–1-1-onto→𝑡 ↔ 𝑓:(0..^3)–1-1-onto→𝑇))
76	75	anbi1d 632	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → ((𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ↔ (𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))))
77	76	exbidv 1923	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ↔ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))))
78	77	elrab 3647	. . . . . . . . 9 ⊢ (𝑇 ∈ {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))} ↔ (𝑇 ∈ 𝒫 𝑉 ∧ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))))
79	74, 78	bitrdi 287	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑉 → (𝑇 ∈ (GrTriangles‘𝐺) ↔ (𝑇 ∈ 𝒫 𝑉 ∧ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)))))
80	79	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑇 ∈ (GrTriangles‘𝐺) ↔ (𝑇 ∈ 𝒫 𝑉 ∧ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)))))
81	80	adantr 480	. . . . . 6 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))) → (𝑇 ∈ (GrTriangles‘𝐺) ↔ (𝑇 ∈ 𝒫 𝑉 ∧ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)))))
82	70, 81	mpbird 257	. . . . 5 ⊢ (((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))) → 𝑇 ∈ (GrTriangles‘𝐺))
83	82	ex 412	. . . 4 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → 𝑇 ∈ (GrTriangles‘𝐺)))
84	83	rexlimdvva 3194	. . 3 ⊢ (𝑥 ∈ 𝑉 → (∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → 𝑇 ∈ (GrTriangles‘𝐺)))
85	84	rexlimiv 3131	. 2 ⊢ (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → 𝑇 ∈ (GrTriangles‘𝐺))
86	3, 85	impbii 209	1 ⊢ (𝑇 ∈ (GrTriangles‘𝐺) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3061  {crab 3400  Vcvv 3441   ∪ cun 3900  ifcif 4480  𝒫 cpw 4555  {csn 4581  {cpr 4583  {ctp 4585   ↦ cmpt 5180  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7360  0cc0 11030  1c1 11031  2c2 12204  3c3 12205  ..^cfzo 13574  ♯chash 14257  Vtxcvtx 29073  Edgcedg 29124  GrTrianglescgrtri 48250

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 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-3o 8401  df-oadd 8403  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9817  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-3 12213  df-n0 12406  df-xnn0 12479  df-z 12493  df-uz 12756  df-fz 13428  df-fzo 13575  df-hash 14258  df-grtri 48251

This theorem is referenced by:  cycl3grtri  48260  grimgrtri  48262  usgrgrtrirex  48263  grlimgrtri  48316  usgrexmpl1tri  48338