Theorem constr3trllem5 26182

Description: Lemma for constr3trl 26187. (Contributed by Alexander van der Vekens, 12-Nov-2017.)

Hypotheses

Ref	Expression
constr3cycl.f	⊢ 𝐹 = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩}
constr3cycl.p	⊢ 𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})

Assertion

Ref	Expression
constr3trllem5	⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})

Distinct variable groups:   𝑘,𝐸   𝑘,𝐹   𝑃,𝑘

Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)   𝐶(𝑘)   𝑉(𝑘)

Proof of Theorem constr3trllem5

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgraedgrnv 25906	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
2	1	ex 449	. . . . . 6 ⊢ (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))
3		usgraedgrnv 25906	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
4	3	ex 449	. . . . . 6 ⊢ (𝑉 USGrph 𝐸 → ({𝐵, 𝐶} ∈ ran 𝐸 → (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
5		usgraedgrnv 25906	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))
6	5	ex 449	. . . . . 6 ⊢ (𝑉 USGrph 𝐸 → ({𝐶, 𝐴} ∈ ran 𝐸 → (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)))
7	2, 4, 6	3anim123d 1398	. . . . 5 ⊢ (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉))))
8	7	imp 444	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)))
9		simpll 786	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
10		simprl 790	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
11		simprr 792	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉)
12		constr3cycl.f	. . . . . . . 8 ⊢ 𝐹 = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩}
13		constr3cycl.p	. . . . . . . 8 ⊢ 𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})
14	12, 13	constr3lem4 26175	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)))
15	9, 10, 11, 14	syl3anc 1318	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)))
16		usgraf 25875	. . . . . . . . . . . . 13 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
17		f1f1orn 6061	. . . . . . . . . . . . 13 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} → 𝐸:dom 𝐸–1-1-onto→ran 𝐸)
18		f1ocnvfv2 6433	. . . . . . . . . . . . . . 15 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (𝐸‘(◡𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵})
19	18	ex 449	. . . . . . . . . . . . . 14 ⊢ (𝐸:dom 𝐸–1-1-onto→ran 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → (𝐸‘(◡𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵}))
20		f1ocnvfv2 6433	. . . . . . . . . . . . . . 15 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐸‘(◡𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶})
21	20	ex 449	. . . . . . . . . . . . . 14 ⊢ (𝐸:dom 𝐸–1-1-onto→ran 𝐸 → ({𝐵, 𝐶} ∈ ran 𝐸 → (𝐸‘(◡𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}))
22		f1ocnvfv2 6433	. . . . . . . . . . . . . . 15 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (𝐸‘(◡𝐸‘{𝐶, 𝐴})) = {𝐶, 𝐴})
23	22	ex 449	. . . . . . . . . . . . . 14 ⊢ (𝐸:dom 𝐸–1-1-onto→ran 𝐸 → ({𝐶, 𝐴} ∈ ran 𝐸 → (𝐸‘(◡𝐸‘{𝐶, 𝐴})) = {𝐶, 𝐴}))
24	19, 21, 23	3anim123d 1398	. . . . . . . . . . . . 13 ⊢ (𝐸:dom 𝐸–1-1-onto→ran 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ((𝐸‘(◡𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(◡𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶} ∧ (𝐸‘(◡𝐸‘{𝐶, 𝐴})) = {𝐶, 𝐴})))
25	16, 17, 24	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ((𝐸‘(◡𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(◡𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶} ∧ (𝐸‘(◡𝐸‘{𝐶, 𝐴})) = {𝐶, 𝐴})))
26	25	imp 444	. . . . . . . . . . 11 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐸‘(◡𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(◡𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶} ∧ (𝐸‘(◡𝐸‘{𝐶, 𝐴})) = {𝐶, 𝐴}))
27	26	adantl 481	. . . . . . . . . 10 ⊢ ((((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐸‘(◡𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(◡𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶} ∧ (𝐸‘(◡𝐸‘{𝐶, 𝐴})) = {𝐶, 𝐴}))
28	12, 13	constr3lem5 26176	. . . . . . . . . . 11 ⊢ ((𝐹‘0) = (◡𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (◡𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (◡𝐸‘{𝐶, 𝐴}))
29		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘0) = (◡𝐸‘{𝐴, 𝐵}) → (𝐸‘(𝐹‘0)) = (𝐸‘(◡𝐸‘{𝐴, 𝐵})))
30	29	3ad2ant1 1075	. . . . . . . . . . . . 13 ⊢ (((𝐹‘0) = (◡𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (◡𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (◡𝐸‘{𝐶, 𝐴})) → (𝐸‘(𝐹‘0)) = (𝐸‘(◡𝐸‘{𝐴, 𝐵})))
31	30	eqeq1d 2612	. . . . . . . . . . . 12 ⊢ (((𝐹‘0) = (◡𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (◡𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (◡𝐸‘{𝐶, 𝐴})) → ((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ↔ (𝐸‘(◡𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵}))
32		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘1) = (◡𝐸‘{𝐵, 𝐶}) → (𝐸‘(𝐹‘1)) = (𝐸‘(◡𝐸‘{𝐵, 𝐶})))
33	32	3ad2ant2 1076	. . . . . . . . . . . . 13 ⊢ (((𝐹‘0) = (◡𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (◡𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (◡𝐸‘{𝐶, 𝐴})) → (𝐸‘(𝐹‘1)) = (𝐸‘(◡𝐸‘{𝐵, 𝐶})))
34	33	eqeq1d 2612	. . . . . . . . . . . 12 ⊢ (((𝐹‘0) = (◡𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (◡𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (◡𝐸‘{𝐶, 𝐴})) → ((𝐸‘(𝐹‘1)) = {𝐵, 𝐶} ↔ (𝐸‘(◡𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}))
35		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘2) = (◡𝐸‘{𝐶, 𝐴}) → (𝐸‘(𝐹‘2)) = (𝐸‘(◡𝐸‘{𝐶, 𝐴})))
36	35	3ad2ant3 1077	. . . . . . . . . . . . 13 ⊢ (((𝐹‘0) = (◡𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (◡𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (◡𝐸‘{𝐶, 𝐴})) → (𝐸‘(𝐹‘2)) = (𝐸‘(◡𝐸‘{𝐶, 𝐴})))
37	36	eqeq1d 2612	. . . . . . . . . . . 12 ⊢ (((𝐹‘0) = (◡𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (◡𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (◡𝐸‘{𝐶, 𝐴})) → ((𝐸‘(𝐹‘2)) = {𝐶, 𝐴} ↔ (𝐸‘(◡𝐸‘{𝐶, 𝐴})) = {𝐶, 𝐴}))
38	31, 34, 37	3anbi123d 1391	. . . . . . . . . . 11 ⊢ (((𝐹‘0) = (◡𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (◡𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (◡𝐸‘{𝐶, 𝐴})) → (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶} ∧ (𝐸‘(𝐹‘2)) = {𝐶, 𝐴}) ↔ ((𝐸‘(◡𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(◡𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶} ∧ (𝐸‘(◡𝐸‘{𝐶, 𝐴})) = {𝐶, 𝐴})))
39	28, 38	ax-mp 5	. . . . . . . . . 10 ⊢ (((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶} ∧ (𝐸‘(𝐹‘2)) = {𝐶, 𝐴}) ↔ ((𝐸‘(◡𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(◡𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶} ∧ (𝐸‘(◡𝐸‘{𝐶, 𝐴})) = {𝐶, 𝐴}))
40	27, 39	sylibr 223	. . . . . . . . 9 ⊢ ((((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶} ∧ (𝐸‘(𝐹‘2)) = {𝐶, 𝐴}))
41		simpll 786	. . . . . . . . . . . . 13 ⊢ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → (𝑃‘0) = 𝐴)
42		simplr 788	. . . . . . . . . . . . 13 ⊢ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → (𝑃‘1) = 𝐵)
43	41, 42	preq12d 4220	. . . . . . . . . . . 12 ⊢ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → {(𝑃‘0), (𝑃‘1)} = {𝐴, 𝐵})
44	43	eqeq2d 2620	. . . . . . . . . . 11 ⊢ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ↔ (𝐸‘(𝐹‘0)) = {𝐴, 𝐵}))
45		simprl 790	. . . . . . . . . . . . 13 ⊢ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → (𝑃‘2) = 𝐶)
46	42, 45	preq12d 4220	. . . . . . . . . . . 12 ⊢ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → {(𝑃‘1), (𝑃‘2)} = {𝐵, 𝐶})
47	46	eqeq2d 2620	. . . . . . . . . . 11 ⊢ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → ((𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ↔ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶}))
48		simprr 792	. . . . . . . . . . . . 13 ⊢ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → (𝑃‘3) = 𝐴)
49	45, 48	preq12d 4220	. . . . . . . . . . . 12 ⊢ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → {(𝑃‘2), (𝑃‘3)} = {𝐶, 𝐴})
50	49	eqeq2d 2620	. . . . . . . . . . 11 ⊢ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ↔ (𝐸‘(𝐹‘2)) = {𝐶, 𝐴}))
51	44, 47, 50	3anbi123d 1391	. . . . . . . . . 10 ⊢ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}) ↔ ((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶} ∧ (𝐸‘(𝐹‘2)) = {𝐶, 𝐴})))
52	51	ad2antrr 758	. . . . . . . . 9 ⊢ ((((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}) ↔ ((𝐸‘(𝐹‘0)) = {𝐴, 𝐵} ∧ (𝐸‘(𝐹‘1)) = {𝐵, 𝐶} ∧ (𝐸‘(𝐹‘2)) = {𝐶, 𝐴})))
53	40, 52	mpbird 246	. . . . . . . 8 ⊢ ((((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}))
54	53	ex 449	. . . . . . 7 ⊢ (((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) → ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)})))
55	54	ex 449	. . . . . 6 ⊢ ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}))))
56	15, 55	mpcom 37	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)})))
57	56	3adant3 1074	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)) → ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)})))
58	8, 57	mpcom 37	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}))
59		0z 11265	. . . 4 ⊢ 0 ∈ ℤ
60		1z 11284	. . . 4 ⊢ 1 ∈ ℤ
61		2z 11286	. . . 4 ⊢ 2 ∈ ℤ
62		fveq2 6103	. . . . . . 7 ⊢ (𝑘 = 0 → (𝐹‘𝑘) = (𝐹‘0))
63	62	fveq2d 6107	. . . . . 6 ⊢ (𝑘 = 0 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘0)))
64		fveq2 6103	. . . . . . 7 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0))
65		oveq1 6556	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
66		0p1e1 11009	. . . . . . . . 9 ⊢ (0 + 1) = 1
67	65, 66	syl6eq 2660	. . . . . . . 8 ⊢ (𝑘 = 0 → (𝑘 + 1) = 1)
68	67	fveq2d 6107	. . . . . . 7 ⊢ (𝑘 = 0 → (𝑃‘(𝑘 + 1)) = (𝑃‘1))
69	64, 68	preq12d 4220	. . . . . 6 ⊢ (𝑘 = 0 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘0), (𝑃‘1)})
70	63, 69	eqeq12d 2625	. . . . 5 ⊢ (𝑘 = 0 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
71		fveq2 6103	. . . . . . 7 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
72	71	fveq2d 6107	. . . . . 6 ⊢ (𝑘 = 1 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘1)))
73		fveq2 6103	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1))
74		oveq1 6556	. . . . . . . . 9 ⊢ (𝑘 = 1 → (𝑘 + 1) = (1 + 1))
75		1p1e2 11011	. . . . . . . . 9 ⊢ (1 + 1) = 2
76	74, 75	syl6eq 2660	. . . . . . . 8 ⊢ (𝑘 = 1 → (𝑘 + 1) = 2)
77	76	fveq2d 6107	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑃‘(𝑘 + 1)) = (𝑃‘2))
78	73, 77	preq12d 4220	. . . . . 6 ⊢ (𝑘 = 1 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘1), (𝑃‘2)})
79	72, 78	eqeq12d 2625	. . . . 5 ⊢ (𝑘 = 1 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
80		fveq2 6103	. . . . . . 7 ⊢ (𝑘 = 2 → (𝐹‘𝑘) = (𝐹‘2))
81	80	fveq2d 6107	. . . . . 6 ⊢ (𝑘 = 2 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘2)))
82		fveq2 6103	. . . . . . 7 ⊢ (𝑘 = 2 → (𝑃‘𝑘) = (𝑃‘2))
83		oveq1 6556	. . . . . . . . 9 ⊢ (𝑘 = 2 → (𝑘 + 1) = (2 + 1))
84		2p1e3 11028	. . . . . . . . 9 ⊢ (2 + 1) = 3
85	83, 84	syl6eq 2660	. . . . . . . 8 ⊢ (𝑘 = 2 → (𝑘 + 1) = 3)
86	85	fveq2d 6107	. . . . . . 7 ⊢ (𝑘 = 2 → (𝑃‘(𝑘 + 1)) = (𝑃‘3))
87	82, 86	preq12d 4220	. . . . . 6 ⊢ (𝑘 = 2 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘2), (𝑃‘3)})
88	81, 87	eqeq12d 2625	. . . . 5 ⊢ (𝑘 = 2 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}))
89	70, 79, 88	raltpg 4183	. . . 4 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 2 ∈ ℤ) → (∀𝑘 ∈ {0, 1, 2} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)})))
90	59, 60, 61, 89	mp3an 1416	. . 3 ⊢ (∀𝑘 ∈ {0, 1, 2} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}))
91	58, 90	sylibr 223	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∀𝑘 ∈ {0, 1, 2} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
92	12, 13	constr3lem2 26174	. . . . . 6 ⊢ (#‘𝐹) = 3
93	92	oveq2i 6560	. . . . 5 ⊢ (0..^(#‘𝐹)) = (0..^3)
94		fzo0to3tp 12421	. . . . 5 ⊢ (0..^3) = {0, 1, 2}
95	93, 94	eqtri 2632	. . . 4 ⊢ (0..^(#‘𝐹)) = {0, 1, 2}
96	95	a1i 11	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (0..^(#‘𝐹)) = {0, 1, 2})
97	96	raleqdv 3121	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1, 2} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
98	91, 97	mpbird 246	1 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ∖ cdif 3537   ∪ cun 3538  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127  {ctp 4129  ⟨cop 4131   class class class wbr 4583  ^◡ccnv 5037  dom cdm 5038  ran crn 5039  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  3c3 10948  ℤcz 11254  ..^cfzo 12334  #chash 12979   USGrph cusg 25859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-usgra 25862

This theorem is referenced by:  constr3trl  26187