Theorem isgrtri 48570

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 48568	. 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 4589	. . . . . . . . . 10 ⊢ {𝑥, 𝑦, 𝑧} = ({𝑥, 𝑦} ∪ {𝑧})
5		prelpwi 5416	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → {𝑥, 𝑦} ∈ 𝒫 𝑉)
6		snelpwi 5413	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑉 → {𝑧} ∈ 𝒫 𝑉)
7	5, 6	anim12i 622	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑉) → ({𝑥, 𝑦} ∈ 𝒫 𝑉 ∧ {𝑧} ∈ 𝒫 𝑉))
8	7	anasss 470	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ({𝑥, 𝑦} ∈ 𝒫 𝑉 ∧ {𝑧} ∈ 𝒫 𝑉))
9		pwuncl 7755	. . . . . . . . . . 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 2868	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → {𝑥, 𝑦, 𝑧} ∈ 𝒫 𝑉)
12	11	adantr 484	. . . . . . . 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 2852	. . . . . . . . . 10 ⊢ (𝑇 = {𝑥, 𝑦, 𝑧} → (𝑇 ∈ 𝒫 𝑉 ↔ {𝑥, 𝑦, 𝑧} ∈ 𝒫 𝑉))
14	13	3ad2ant1 1147	. . . . . . . . 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 485	. . . . . . . 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 259	. . . . . . 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 7431	. . . . . . . . . 10 ⊢ (0..^3) ∈ V
18	17	mptex 7209	. . . . . . . . 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 1107	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) ↔ (𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)))
21	20	biimpri 230	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉))
22		fveq2 6869	. . . . . . . . . . . . . 14 ⊢ (𝑇 = {𝑥, 𝑦, 𝑧} → (♯‘𝑇) = (♯‘{𝑥, 𝑦, 𝑧}))
23	22	eqcomd 2770	. . . . . . . . . . . . 13 ⊢ (𝑇 = {𝑥, 𝑦, 𝑧} → (♯‘{𝑥, 𝑦, 𝑧}) = (♯‘𝑇))
24	23	3ad2ant1 1147	. . . . . . . . . . . 12 ⊢ ((𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → (♯‘{𝑥, 𝑦, 𝑧}) = (♯‘𝑇))
25		simp2 1151	. . . . . . . . . . . 12 ⊢ ((𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → (♯‘𝑇) = 3)
26	24, 25	eqtrd 2799	. . . . . . . . . . 11 ⊢ ((𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)) → (♯‘{𝑥, 𝑦, 𝑧}) = 3)
27		eqid 2764	. . . . . . . . . . . 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 2764	. . . . . . . . . . . 12 ⊢ {𝑥, 𝑦, 𝑧} = {𝑥, 𝑦, 𝑧}
29	27, 28	tpf1o 14516	. . . . . . . . . . 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 605	. . . . . . . . . 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 6798	. . . . . . . . . . . 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 1147	. . . . . . . . . . 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 485	. . . . . . . . . 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 259	. . . . . . . . 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 14511	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑉 → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0) = 𝑥)
36	35	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘0) = 𝑥)
37	27	tpf1ofv1 14512	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ 𝑉 → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1) = 𝑦)
38	37	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘1) = 𝑦)
39	38	adantl 485	. . . . . . . . . . . . . . . 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 4702	. . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . 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 2849	. . . . . . . . . . . . 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 14513	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝑉 → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2) = 𝑧)
44	43	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝑖 ∈ (0..^3) ↦ if(𝑖 = 0, 𝑥, if(𝑖 = 1, 𝑦, 𝑧)))‘2) = 𝑧)
45	44	adantl 485	. . . . . . . . . . . . . . . 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 4702	. . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . 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 2849	. . . . . . . . . . . . 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 4702	. . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . 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 2849	. . . . . . . . . . . . 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 1459	. . . . . . . . . . . 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 231	. . . . . . . . . . 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 1364	. . . . . . . . 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 519	. . . . . . . 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 6796	. . . . . . . . 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 6868	. . . . . . . . . . . 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 6868	. . . . . . . . . . . 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 4702	. . . . . . . . . . 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 2849	. . . . . . . . . 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 6868	. . . . . . . . . . . 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 4702	. . . . . . . . . . 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 2849	. . . . . . . . . 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 4702	. . . . . . . . . . 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 2849	. . . . . . . . . 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 1459	. . . . . . . . 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 641	. . . . . . . 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 3559	. . . . . . 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 519	. . . . . 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 29205	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑉 → 𝐺 ∈ V)
72	1, 2	grtri 48567	. . . . . . . . . . 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 2850	. . . . . . . . . 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 6798	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (𝑓:(0..^3)–1-1-onto→𝑡 ↔ 𝑓:(0..^3)–1-1-onto→𝑇))
76	75	anbi1d 640	. . . . . . . . . . 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 1943	. . . . . . . . . 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 3652	. . . . . . . . 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 289	. . . . . . . 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 484	. . . . . . 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 484	. . . . . 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 259	. . . . 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 416	. . . 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 3221	. . 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 3158	. 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 211	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 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562  ∃wex 1801   ∈ wcel 2144  ∃wrex 3088  {crab 3416  Vcvv 3456   ∪ cun 3904  ifcif 4482  𝒫 cpw 4557  {csn 4584  {cpr 4586  {ctp 4588   ↦ cmpt 5183  –1-1-onto→wf1o 6522  ‘cfv 6523  (class class class)co 7398  0cc0 11075  1c1 11076  2c2 12274  3c3 12275  ..^cfzo 13661  ♯chash 14345  Vtxcvtx 29199  Edgcedg 29250  GrTrianglescgrtri 48564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-3o 8441  df-oadd 8443  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-dju 9861  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-2 12282  df-3 12283  df-n0 12484  df-xnn0 12557  df-z 12571  df-uz 12842  df-fz 13515  df-fzo 13662  df-hash 14346  df-grtri 48565

This theorem is referenced by:  cycl3grtri  48574  grimgrtri  48576  usgrgrtrirex  48577  grlimgrtri  48630  usgrexmpl1tri  48652