Theorem grimgrtri 47852

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 2735	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2735	. . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
4	2, 3	grtriprop 47846	. . . 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 2735	. . . . . . . . . 10 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
8	2, 7	grimf1o 47808	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
9		f1of1 6848	. . . . . . . . 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 777	. . . . . . . . 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 773	. . . . . . . . . 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 769	. . . . . . . 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 1127	. . . . . . 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 1147	. . . . . . . 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 47851	. . . . . . . 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 837	. . . . . 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 2735	. . . . . . . . . . . . . . 15 ⊢ (Edg‘𝐻) = (Edg‘𝐻)
27	2, 3, 26	grimedg 47841	. . . . . . . . . . . . . 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 47841	. . . . . . . . . . . . . 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 47841	. . . . . . . . . . . . . 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 1435	. . . . . . . . . . . . 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 6850	. . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → 𝑎 ∈ (Vtx‘𝐺))
34		simprlr 780	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → 𝑏 ∈ (Vtx‘𝐺))
35		fnimapr 6992	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ 𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) → (𝐹 “ {𝑎, 𝑏}) = {(𝐹‘𝑎), (𝐹‘𝑏)})
36	32, 33, 34, 35	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → (𝐹 “ {𝑎, 𝑏}) = {(𝐹‘𝑎), (𝐹‘𝑏)})
37	36	eleq1d 2824	. . . . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → 𝑐 ∈ (Vtx‘𝐺))
41		fnimapr 6992	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ 𝑎 ∈ (Vtx‘𝐺) ∧ 𝑐 ∈ (Vtx‘𝐺)) → (𝐹 “ {𝑎, 𝑐}) = {(𝐹‘𝑎), (𝐹‘𝑐)})
42	32, 33, 40, 41	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → (𝐹 “ {𝑎, 𝑐}) = {(𝐹‘𝑎), (𝐹‘𝑐)})
43	42	eleq1d 2824	. . . . . . . . . . . . . . . . . . . 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 6992	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺) ∧ 𝑐 ∈ (Vtx‘𝐺)) → (𝐹 “ {𝑏, 𝑐}) = {(𝐹‘𝑏), (𝐹‘𝑐)})
47	32, 34, 40, 46	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 Fn (Vtx‘𝐺) ∧ ((𝑎 ∈ (Vtx‘𝐺) ∧ 𝑏 ∈ (Vtx‘𝐺)) ∧ 𝑐 ∈ (Vtx‘𝐺))) → (𝐹 “ {𝑏, 𝑐}) = {(𝐹‘𝑏), (𝐹‘𝑐)})
48	47	eleq1d 2824	. . . . . . . . . . . . . . . . . . . 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 1442	. . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . 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 1347	. . . . . . . . . 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 1370	. . . . . . . 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 4747	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹‘𝑎) → {𝑥, 𝑦, 𝑧} = {(𝐹‘𝑎), 𝑦, 𝑧})
64	63	eqeq2d 2746	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝐹 “ 𝑇) = {𝑥, 𝑦, 𝑧} ↔ (𝐹 “ 𝑇) = {(𝐹‘𝑎), 𝑦, 𝑧}))
65		preq1 4738	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐹‘𝑎) → {𝑥, 𝑦} = {(𝐹‘𝑎), 𝑦})
66	65	eleq1d 2824	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹‘𝑎) → ({𝑥, 𝑦} ∈ (Edg‘𝐻) ↔ {(𝐹‘𝑎), 𝑦} ∈ (Edg‘𝐻)))
67		preq1 4738	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐹‘𝑎) → {𝑥, 𝑧} = {(𝐹‘𝑎), 𝑧})
68	67	eleq1d 2824	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹‘𝑎) → ({𝑥, 𝑧} ∈ (Edg‘𝐻) ↔ {(𝐹‘𝑎), 𝑧} ∈ (Edg‘𝐻)))
69	66, 68	3anbi12d 1436	. . . . . . . . . 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 1437	. . . . . . . . 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 4748	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑏) → {(𝐹‘𝑎), 𝑦, 𝑧} = {(𝐹‘𝑎), (𝐹‘𝑏), 𝑧})
72	71	eqeq2d 2746	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹 “ 𝑇) = {(𝐹‘𝑎), 𝑦, 𝑧} ↔ (𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), 𝑧}))
73		preq2 4739	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑏) → {(𝐹‘𝑎), 𝑦} = {(𝐹‘𝑎), (𝐹‘𝑏)})
74	73	eleq1d 2824	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑏) → ({(𝐹‘𝑎), 𝑦} ∈ (Edg‘𝐻) ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ (Edg‘𝐻)))
75		preq1 4738	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑏) → {𝑦, 𝑧} = {(𝐹‘𝑏), 𝑧})
76	75	eleq1d 2824	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑏) → ({𝑦, 𝑧} ∈ (Edg‘𝐻) ↔ {(𝐹‘𝑏), 𝑧} ∈ (Edg‘𝐻)))
77	74, 76	3anbi13d 1437	. . . . . . . . . 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 1437	. . . . . . . . 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 4749	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑐) → {(𝐹‘𝑎), (𝐹‘𝑏), 𝑧} = {(𝐹‘𝑎), (𝐹‘𝑏), (𝐹‘𝑐)})
80	79	eqeq2d 2746	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹‘𝑐) → ((𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), 𝑧} ↔ (𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), (𝐹‘𝑐)}))
81		preq2 4739	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘𝑐) → {(𝐹‘𝑎), 𝑧} = {(𝐹‘𝑎), (𝐹‘𝑐)})
82	81	eleq1d 2824	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑐) → ({(𝐹‘𝑎), 𝑧} ∈ (Edg‘𝐻) ↔ {(𝐹‘𝑎), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))
83		preq2 4739	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘𝑐) → {(𝐹‘𝑏), 𝑧} = {(𝐹‘𝑏), (𝐹‘𝑐)})
84	83	eleq1d 2824	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑐) → ({(𝐹‘𝑏), 𝑧} ∈ (Edg‘𝐻) ↔ {(𝐹‘𝑏), (𝐹‘𝑐)} ∈ (Edg‘𝐻)))
85	82, 84	3anbi23d 1438	. . . . . . . . . 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 1437	. . . . . . . . 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 3639	. . . . . . . 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 1353	. . . . . . 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 3155	. . . 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 3211	. . 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 47848	. 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))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∃wrex 3068   ⊆ wss 3963  {cpr 4633  {ctp 4635   “ cima 5692   Fn wfn 6558  –1-1→wf1 6560  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431  3c3 12320  ♯chash 14366  Vtxcvtx 29028  Edgcedg 29079  UHGraphcuhgr 29088   GraphIso cgrim 47799  GrTrianglescgrtri 47842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-3o 8507  df-oadd 8509  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-edg 29080  df-uhgr 29090  df-grim 47802  df-grtri 47843

This theorem is referenced by:  usgrexmpl12ngric  47933