grimgrtri - Mathbox for Alexander van der Vekens

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Theorem grimgrtri 47959

Description: Graph isomorphisms map triangles onto triangles. (Contributed by AV, 27-Jul-2025.) (Proof shortened by AV, 24-Aug-2025.)

**Hypotheses**
Ref	Expression
grimgrtri.g	⊢ (𝜑 → 𝐺 ∈ UHGraph)
grimgrtri.h	⊢ (𝜑 → 𝐻 ∈ UHGraph)
grimgrtri.n	⊢ (𝜑 → 𝐹 ∈ (𝐺 GraphIso 𝐻))
grimgrtri.t	⊢ (𝜑 → 𝑇 ∈ (GrTriangles‘𝐺))

**Assertion**
Ref	Expression
grimgrtri	⊢ (𝜑 → (𝐹 “ 𝑇) ∈ (GrTriangles‘𝐻))

Proof of Theorem grimgrtri Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grimgrtri.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (GrTriangles‘𝐺))
2		eqid 2730	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2730	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
4	2, 3	grtriprop 47951	. . . 4 ⊢ (𝑇 ∈ (GrTriangles‘𝐺) → ∃𝑎 ∈ (Vtx‘𝐺)∃𝑏 ∈ (Vtx‘𝐺)∃𝑐 ∈ (Vtx‘𝐺)(𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))))
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ (Vtx‘𝐺)∃𝑏 ∈ (Vtx‘𝐺)∃𝑐 ∈ (Vtx‘𝐺)(𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))))
6		grimgrtri.n	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GraphIso 𝐻))
7		eqid 2730	. . . . . . . . . 10 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
8	2, 7	grimf1o 47894	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
9		f1of1 6758	. . . . . . . . 9 ⊢ (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → 𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
10	6, 8, 9	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
11	10	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺))) ∧ 𝑐 ∈ (Vtx‘𝐺)) ∧ (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))) → 𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
12		simplrl 776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺))) ∧ 𝑐 ∈ (Vtx‘𝐺)) → 𝑎 ∈ (Vtx‘𝐺))
13	12	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺))) ∧ 𝑐 ∈ (Vtx‘𝐺)) ∧ (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))) → 𝑎 ∈ (Vtx‘𝐺))
14		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺))) → 𝑏 ∈ (Vtx‘𝐺))
15	14	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺))) ∧ 𝑐 ∈ (Vtx‘𝐺)) → 𝑏 ∈ (Vtx‘𝐺))
16	15	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺))) ∧ 𝑐 ∈ (Vtx‘𝐺)) ∧ (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))) → 𝑏 ∈ (Vtx‘𝐺))
17		simplr 768	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺))) ∧ 𝑐 ∈ (Vtx‘𝐺)) ∧ (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))) → 𝑐 ∈ (Vtx‘𝐺))
18	13, 16, 17	3jca 1128	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺))) ∧ 𝑐 ∈ (Vtx‘𝐺)) ∧ (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))) → (𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺) ∧ 𝑐 ∈ (Vtx‘𝐺)))
19		3simpa 1148	. . . . . . . 8 ⊢ ((𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))) → (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3))
20	19	adantl 481	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺))) ∧ 𝑐 ∈ (Vtx‘𝐺)) ∧ (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))) → (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3))
21		grtrimap 47958	. . . . . . . 8 ⊢ (𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻) → (((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺) ∧ 𝑐 ∈ (Vtx‘𝐺)) ∧ (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3)) → (((𝐹‘𝑎) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑏) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑐) ∈ (Vtx‘𝐻)) ∧ (𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), (𝐹‘𝑐)} ∧ (♯‘(𝐹 “ 𝑇)) = 3)))
22	21	imp 406	. . . . . . 7 ⊢ ((𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺) ∧ 𝑐 ∈ (Vtx‘𝐺)) ∧ (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3))) → (((𝐹‘𝑎) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑏) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑐) ∈ (Vtx‘𝐻)) ∧ (𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), (𝐹‘𝑐)} ∧ (♯‘(𝐹 “ 𝑇)) = 3))
23	11, 18, 20, 22	syl12anc 836	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺))) ∧ 𝑐 ∈ (Vtx‘𝐺)) ∧ (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))) → (((𝐹‘𝑎) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑏) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑐) ∈ (Vtx‘𝐻)) ∧ (𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), (𝐹‘𝑐)} ∧ (♯‘(𝐹 “ 𝑇)) = 3))
24		grimgrtri.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
25		grimgrtri.h	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ UHGraph)
26		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (Edg‘𝐻) = (Edg‘𝐻)
27	2, 3, 26	grimedg 47945	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → ({𝑎, 𝑏} ∈ (Edg‘𝐺) ↔ ((𝐹 “ {𝑎, 𝑏}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑏} ⊆ (Vtx‘𝐺))))
28	2, 3, 26	grimedg 47945	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → ({𝑎, 𝑐} ∈ (Edg‘𝐺) ↔ ((𝐹 “ {𝑎, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑐} ⊆ (Vtx‘𝐺))))
29	2, 3, 26	grimedg 47945	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → ({𝑏, 𝑐} ∈ (Edg‘𝐺) ↔ ((𝐹 “ {𝑏, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑏, 𝑐} ⊆ (Vtx‘𝐺))))
30	27, 28, 29	3anbi123d 1438	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)) ↔ (((𝐹 “ {𝑎, 𝑏}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑏} ⊆ (Vtx‘𝐺)) ∧ ((𝐹 “ {𝑎, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑐} ⊆ (Vtx‘𝐺)) ∧ ((𝐹 “ {𝑏, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑏, 𝑐} ⊆ (Vtx‘𝐺)))))
31		f1ofn 6760	. . . . . . . . . . . . . . 15 ⊢ (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → 𝐹 Fn (Vtx‘𝐺))
32		simpl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → 𝐹 Fn (Vtx‘𝐺))
33		simprll 778	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → 𝑎 ∈ (Vtx‘𝐺))
34		simprlr 779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → 𝑏 ∈ (Vtx‘𝐺))
35		fnimapr 6900	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ 𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) → (𝐹 “ {𝑎, 𝑏}) = {(𝐹‘𝑎), (𝐹‘𝑏)})
36	32, 33, 34, 35	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → (𝐹 “ {𝑎, 𝑏}) = {(𝐹‘𝑎), (𝐹‘𝑏)})
37	36	eleq1d 2814	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → ((𝐹 “ {𝑎, 𝑏}) ∈ (Edg‘𝐻) ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻)))
38	37	biimpd 229	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → ((𝐹 “ {𝑎, 𝑏}) ∈ (Edg‘𝐻) → {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻)))
39	38	adantrd 491	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → (((𝐹 “ {𝑎, 𝑏}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑏} ⊆ (Vtx‘𝐺)) → {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻)))
40		simprr 772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → 𝑐 ∈ (Vtx‘𝐺))
41		fnimapr 6900	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ 𝑎 ∈ (Vtx‘𝐺) ∧ 𝑐 ∈ (Vtx‘𝐺)) → (𝐹 “ {𝑎, 𝑐}) = {(𝐹‘𝑎), (𝐹‘𝑐)})
42	32, 33, 40, 41	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → (𝐹 “ {𝑎, 𝑐}) = {(𝐹‘𝑎), (𝐹‘𝑐)})
43	42	eleq1d 2814	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → ((𝐹 “ {𝑎, 𝑐}) ∈ (Edg‘𝐻) ↔ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))
44	43	biimpd 229	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → ((𝐹 “ {𝑎, 𝑐}) ∈ (Edg‘𝐻) → {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))
45	44	adantrd 491	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → (((𝐹 “ {𝑎, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑐} ⊆ (Vtx‘𝐺)) → {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))
46		fnimapr 6900	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺) ∧ 𝑐 ∈ (Vtx‘𝐺)) → (𝐹 “ {𝑏, 𝑐}) = {(𝐹‘𝑏), (𝐹‘𝑐)})
47	32, 34, 40, 46	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → (𝐹 “ {𝑏, 𝑐}) = {(𝐹‘𝑏), (𝐹‘𝑐)})
48	47	eleq1d 2814	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → ((𝐹 “ {𝑏, 𝑐}) ∈ (Edg‘𝐻) ↔ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))
49	48	biimpd 229	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → ((𝐹 “ {𝑏, 𝑐}) ∈ (Edg‘𝐻) → {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))
50	49	adantrd 491	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → (((𝐹 “ {𝑏, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑏, 𝑐} ⊆ (Vtx‘𝐺)) → {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))
51	39, 45, 50	3anim123d 1445	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → ((((𝐹 “ {𝑎, 𝑏}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑏} ⊆ (Vtx‘𝐺)) ∧ ((𝐹 “ {𝑎, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑐} ⊆ (Vtx‘𝐺)) ∧ ((𝐹 “ {𝑏, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑏, 𝑐} ⊆ (Vtx‘𝐺))) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻))))
52	51	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 Fn (Vtx‘𝐺) → (((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺)) → ((((𝐹 “ {𝑎, 𝑏}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑏} ⊆ (Vtx‘𝐺)) ∧ ((𝐹 “ {𝑎, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑐} ⊆ (Vtx‘𝐺)) ∧ ((𝐹 “ {𝑏, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑏, 𝑐} ⊆ (Vtx‘𝐺))) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))))
53	52	com23 86	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn (Vtx‘𝐺) → ((((𝐹 “ {𝑎, 𝑏}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑏} ⊆ (Vtx‘𝐺)) ∧ ((𝐹 “ {𝑎, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑐} ⊆ (Vtx‘𝐺)) ∧ ((𝐹 “ {𝑏, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑏, 𝑐} ⊆ (Vtx‘𝐺))) → (((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺)) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))))
54	8, 31, 53	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → ((((𝐹 “ {𝑎, 𝑏}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑏} ⊆ (Vtx‘𝐺)) ∧ ((𝐹 “ {𝑎, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑐} ⊆ (Vtx‘𝐺)) ∧ ((𝐹 “ {𝑏, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑏, 𝑐} ⊆ (Vtx‘𝐺))) → (((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺)) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))))
55	54	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → ((((𝐹 “ {𝑎, 𝑏}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑏} ⊆ (Vtx‘𝐺)) ∧ ((𝐹 “ {𝑎, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑎, 𝑐} ⊆ (Vtx‘𝐺)) ∧ ((𝐹 “ {𝑏, 𝑐}) ∈ (Edg‘𝐻) ∧ {𝑏, 𝑐} ⊆ (Vtx‘𝐺))) → (((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺)) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))))
56	30, 55	sylbid 240	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)) → (((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺)) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))))
57	56	2a1d 26	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝑇 = {𝑎, 𝑏, 𝑐} → ((♯‘𝑇) = 3 → (({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)) → (((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺)) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))))))
58	57	3impd 1349	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → ((𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))) → (((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺)) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))))
59	58	com23 86	. . . . . . . . 9 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺)) → ((𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))))
60	24, 25, 6, 59	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺)) → ((𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))))
61	60	impl 455	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺))) ∧ 𝑐 ∈ (Vtx‘𝐺)) → ((𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻))))
62	61	imp 406	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺))) ∧ 𝑐 ∈ (Vtx‘𝐺)) ∧ (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))
63		tpeq1 4693	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹‘𝑎) → {𝑥, 𝑦, 𝑧} = {(𝐹‘𝑎), 𝑦, 𝑧})
64	63	eqeq2d 2741	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝐹 “ 𝑇) = {𝑥, 𝑦, 𝑧} ↔ (𝐹 “ 𝑇) = {(𝐹‘𝑎), 𝑦, 𝑧}))
65		preq1 4684	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐹‘𝑎) → {𝑥, 𝑦} = {(𝐹‘𝑎), 𝑦})
66	65	eleq1d 2814	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹‘𝑎) → ({𝑥, 𝑦} ∈ (Edg‘𝐻) ↔ {(𝐹‘𝑎), 𝑦} ∈ (Edg‘𝐻)))
67		preq1 4684	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐹‘𝑎) → {𝑥, 𝑧} = {(𝐹‘𝑎), 𝑧})
68	67	eleq1d 2814	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹‘𝑎) → ({𝑥, 𝑧} ∈ (Edg‘𝐻) ↔ {(𝐹‘𝑎), 𝑧} ∈ (Edg‘𝐻)))
69	66, 68	3anbi12d 1439	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑎) → (({𝑥, 𝑦} ∈ (Edg‘𝐻) ∧ {𝑥, 𝑧} ∈ (Edg‘𝐻) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐻)) ↔ ({(𝐹‘𝑎), 𝑦} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), 𝑧} ∈ (Edg‘𝐻) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐻))))
70	64, 69	3anbi13d 1440	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → (((𝐹 “ 𝑇) = {𝑥, 𝑦, 𝑧} ∧ (♯‘(𝐹 “ 𝑇)) = 3 ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐻) ∧ {𝑥, 𝑧} ∈ (Edg‘𝐻) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐻))) ↔ ((𝐹 “ 𝑇) = {(𝐹‘𝑎), 𝑦, 𝑧} ∧ (♯‘(𝐹 “ 𝑇)) = 3 ∧ ({(𝐹‘𝑎), 𝑦} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), 𝑧} ∈ (Edg‘𝐻) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐻)))))
71		tpeq2 4694	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑏) → {(𝐹‘𝑎), 𝑦, 𝑧} = {(𝐹‘𝑎), (𝐹‘𝑏), 𝑧})
72	71	eqeq2d 2741	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹 “ 𝑇) = {(𝐹‘𝑎), 𝑦, 𝑧} ↔ (𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), 𝑧}))
73		preq2 4685	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑏) → {(𝐹‘𝑎), 𝑦} = {(𝐹‘𝑎), (𝐹‘𝑏)})
74	73	eleq1d 2814	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑏) → ({(𝐹‘𝑎), 𝑦} ∈ (Edg‘𝐻) ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻)))
75		preq1 4684	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑏) → {𝑦, 𝑧} = {(𝐹‘𝑏), 𝑧})
76	75	eleq1d 2814	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑏) → ({𝑦, 𝑧} ∈ (Edg‘𝐻) ↔ {(𝐹‘𝑏), 𝑧} ∈ (Edg‘𝐻)))
77	74, 76	3anbi13d 1440	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → (({(𝐹‘𝑎), 𝑦} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), 𝑧} ∈ (Edg‘𝐻) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐻)) ↔ ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), 𝑧} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), 𝑧} ∈ (Edg‘𝐻))))
78	72, 77	3anbi13d 1440	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹 “ 𝑇) = {(𝐹‘𝑎), 𝑦, 𝑧} ∧ (♯‘(𝐹 “ 𝑇)) = 3 ∧ ({(𝐹‘𝑎), 𝑦} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), 𝑧} ∈ (Edg‘𝐻) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐻))) ↔ ((𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), 𝑧} ∧ (♯‘(𝐹 “ 𝑇)) = 3 ∧ ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), 𝑧} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), 𝑧} ∈ (Edg‘𝐻)))))
79		tpeq3 4695	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑐) → {(𝐹‘𝑎), (𝐹‘𝑏), 𝑧} = {(𝐹‘𝑎), (𝐹‘𝑏), (𝐹‘𝑐)})
80	79	eqeq2d 2741	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹‘𝑐) → ((𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), 𝑧} ↔ (𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), (𝐹‘𝑐)}))
81		preq2 4685	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘𝑐) → {(𝐹‘𝑎), 𝑧} = {(𝐹‘𝑎), (𝐹‘𝑐)})
82	81	eleq1d 2814	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑐) → ({(𝐹‘𝑎), 𝑧} ∈ (Edg‘𝐻) ↔ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))
83		preq2 4685	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘𝑐) → {(𝐹‘𝑏), 𝑧} = {(𝐹‘𝑏), (𝐹‘𝑐)})
84	83	eleq1d 2814	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑐) → ({(𝐹‘𝑏), 𝑧} ∈ (Edg‘𝐻) ↔ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))
85	82, 84	3anbi23d 1441	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹‘𝑐) → (({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), 𝑧} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), 𝑧} ∈ (Edg‘𝐻)) ↔ ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻))))
86	80, 85	3anbi13d 1440	. . . . . . . . 9 ⊢ (𝑧 = (𝐹‘𝑐) → (((𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), 𝑧} ∧ (♯‘(𝐹 “ 𝑇)) = 3 ∧ ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), 𝑧} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), 𝑧} ∈ (Edg‘𝐻))) ↔ ((𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), (𝐹‘𝑐)} ∧ (♯‘(𝐹 “ 𝑇)) = 3 ∧ ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))))
87	70, 78, 86	rspc3ev 3592	. . . . . . . 8 ⊢ ((((𝐹‘𝑎) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑏) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑐) ∈ (Vtx‘𝐻)) ∧ ((𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), (𝐹‘𝑐)} ∧ (♯‘(𝐹 “ 𝑇)) = 3 ∧ ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))) → ∃𝑥 ∈ (Vtx‘𝐻)∃𝑦 ∈ (Vtx‘𝐻)∃𝑧 ∈ (Vtx‘𝐻)((𝐹 “ 𝑇) = {𝑥, 𝑦, 𝑧} ∧ (♯‘(𝐹 “ 𝑇)) = 3 ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐻) ∧ {𝑥, 𝑧} ∈ (Edg‘𝐻) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐻))))
88	87	3exp2 1355	. . . . . . 7 ⊢ (((𝐹‘𝑎) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑏) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑐) ∈ (Vtx‘𝐻)) → ((𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), (𝐹‘𝑐)} → ((♯‘(𝐹 “ 𝑇)) = 3 → (({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)) → ∃𝑥 ∈ (Vtx‘𝐻)∃𝑦 ∈ (Vtx‘𝐻)∃𝑧 ∈ (Vtx‘𝐻)((𝐹 “ 𝑇) = {𝑥, 𝑦, 𝑧} ∧ (♯‘(𝐹 “ 𝑇)) = 3 ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐻) ∧ {𝑥, 𝑧} ∈ (Edg‘𝐻) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐻)))))))
89	88	3imp 1110	. . . . . 6 ⊢ ((((𝐹‘𝑎) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑏) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝑐) ∈ (Vtx‘𝐻)) ∧ (𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), (𝐹‘𝑐)} ∧ (♯‘(𝐹 “ 𝑇)) = 3) → (({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻) ∧ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)) → ∃𝑥 ∈ (Vtx‘𝐻)∃𝑦 ∈ (Vtx‘𝐻)∃𝑧 ∈ (Vtx‘𝐻)((𝐹 “ 𝑇) = {𝑥, 𝑦, 𝑧} ∧ (♯‘(𝐹 “ 𝑇)) = 3 ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐻) ∧ {𝑥, 𝑧} ∈ (Edg‘𝐻) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐻)))))
90	23, 62, 89	sylc 65	. . . . 5 ⊢ ((((𝜑 ∧ (𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺))) ∧ 𝑐 ∈ (Vtx‘𝐺)) ∧ (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺)))) → ∃𝑥 ∈ (Vtx‘𝐻)∃𝑦 ∈ (Vtx‘𝐻)∃𝑧 ∈ (Vtx‘𝐻)((𝐹 “ 𝑇) = {𝑥, 𝑦, 𝑧} ∧ (♯‘(𝐹 “ 𝑇)) = 3 ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐻) ∧ {𝑥, 𝑧} ∈ (Edg‘𝐻) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐻))))
91	90	rexlimdva2 3133	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺))) → (∃𝑐 ∈ (Vtx‘𝐺)(𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))) → ∃𝑥 ∈ (Vtx‘𝐻)∃𝑦 ∈ (Vtx‘𝐻)∃𝑧 ∈ (Vtx‘𝐻)((𝐹 “ 𝑇) = {𝑥, 𝑦, 𝑧} ∧ (♯‘(𝐹 “ 𝑇)) = 3 ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐻) ∧ {𝑥, 𝑧} ∈ (Edg‘𝐻) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐻)))))
92	91	rexlimdvva 3187	. . 3 ⊢ (𝜑 → (∃𝑎 ∈ (Vtx‘𝐺)∃𝑏 ∈ (Vtx‘𝐺)∃𝑐 ∈ (Vtx‘𝐺)(𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3 ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑎, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺))) → ∃𝑥 ∈ (Vtx‘𝐻)∃𝑦 ∈ (Vtx‘𝐻)∃𝑧 ∈ (Vtx‘𝐻)((𝐹 “ 𝑇) = {𝑥, 𝑦, 𝑧} ∧ (♯‘(𝐹 “ 𝑇)) = 3 ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐻) ∧ {𝑥, 𝑧} ∈ (Edg‘𝐻) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐻)))))
93	5, 92	mpd 15	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (Vtx‘𝐻)∃𝑦 ∈ (Vtx‘𝐻)∃𝑧 ∈ (Vtx‘𝐻)((𝐹 “ 𝑇) = {𝑥, 𝑦, 𝑧} ∧ (♯‘(𝐹 “ 𝑇)) = 3 ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐻) ∧ {𝑥, 𝑧} ∈ (Edg‘𝐻) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐻))))
94	7, 26	isgrtri 47953	. 2 ⊢ ((𝐹 “ 𝑇) ∈ (GrTriangles‘𝐻) ↔ ∃𝑥 ∈ (Vtx‘𝐻)∃𝑦 ∈ (Vtx‘𝐻)∃𝑧 ∈ (Vtx‘𝐻)((𝐹 “ 𝑇) = {𝑥, 𝑦, 𝑧} ∧ (♯‘(𝐹 “ 𝑇)) = 3 ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐻) ∧ {𝑥, 𝑧} ∈ (Edg‘𝐻) ∧ {𝑦, 𝑧} ∈ (Edg‘𝐻))))
95	93, 94	sylibr 234	1 ⊢ (𝜑 → (𝐹 “ 𝑇) ∈ (GrTriangles‘𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2110 ∃wrex 3054 ⊆ wss 3900 {cpr 4576 {ctp 4578 “ cima 5617 Fn wfn 6472 –1-1→wf1 6474 –1-1-onto→wf1o 6476 ‘cfv 6477 (class class class)co 7341 3c3 12173 ♯chash 14229 Vtxcvtx 28967 Edgcedg 29018 UHGraphcuhgr 29027 GraphIso cgrim 47885 GrTrianglescgrtri 47947

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-3o 8382 df-oadd 8384 df-er 8617 df-map 8747 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-dju 9786 df-card 9824 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-nn 12118 df-2 12180 df-3 12181 df-n0 12374 df-xnn0 12447 df-z 12461 df-uz 12725 df-fz 13400 df-fzo 13547 df-hash 14230 df-edg 29019 df-uhgr 29029 df-grim 47888 df-grtri 47948

This theorem is referenced by: usgrexmpl12ngric 48048

W3C validator