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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‘(𝐹𝑘)) = {(𝑃𝑘), (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.
StepHypRef Expression
1 usgraedgrnv 25906 . . . . . . 7 ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (𝐴𝑉𝐵𝑉))
21ex 449 . . . . . 6 (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → (𝐴𝑉𝐵𝑉)))
3 usgraedgrnv 25906 . . . . . . 7 ((𝑉 USGrph 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐵𝑉𝐶𝑉))
43ex 449 . . . . . 6 (𝑉 USGrph 𝐸 → ({𝐵, 𝐶} ∈ ran 𝐸 → (𝐵𝑉𝐶𝑉)))
5 usgraedgrnv 25906 . . . . . . 7 ((𝑉 USGrph 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (𝐶𝑉𝐴𝑉))
65ex 449 . . . . . 6 (𝑉 USGrph 𝐸 → ({𝐶, 𝐴} ∈ ran 𝐸 → (𝐶𝑉𝐴𝑉)))
72, 4, 63anim123d 1398 . . . . 5 (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉) ∧ (𝐶𝑉𝐴𝑉))))
87imp 444 . . . 4 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉) ∧ (𝐶𝑉𝐴𝑉)))
9 simpll 786 . . . . . . 7 (((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)) → 𝐴𝑉)
10 simprl 790 . . . . . . 7 (((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)) → 𝐵𝑉)
11 simprr 792 . . . . . . 7 (((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)) → 𝐶𝑉)
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⟩})
1412, 13constr3lem4 26175 . . . . . . 7 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)))
159, 10, 11, 14syl3anc 1318 . . . . . 6 (((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)))
16 usgraf 25875 . . . . . . . . . . . . 13 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
17 f1f1orn 6061 . . . . . . . . . . . . 13 (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} → 𝐸:dom 𝐸1-1-onto→ran 𝐸)
18 f1ocnvfv2 6433 . . . . . . . . . . . . . . 15 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵})
1918ex 449 . . . . . . . . . . . . . 14 (𝐸:dom 𝐸1-1-onto→ran 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → (𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵}))
20 f1ocnvfv2 6433 . . . . . . . . . . . . . . 15 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶})
2120ex 449 . . . . . . . . . . . . . 14 (𝐸:dom 𝐸1-1-onto→ran 𝐸 → ({𝐵, 𝐶} ∈ ran 𝐸 → (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}))
22 f1ocnvfv2 6433 . . . . . . . . . . . . . . 15 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (𝐸‘(𝐸‘{𝐶, 𝐴})) = {𝐶, 𝐴})
2322ex 449 . . . . . . . . . . . . . 14 (𝐸:dom 𝐸1-1-onto→ran 𝐸 → ({𝐶, 𝐴} ∈ ran 𝐸 → (𝐸‘(𝐸‘{𝐶, 𝐴})) = {𝐶, 𝐴}))
2419, 21, 233anim123d 1398 . . . . . . . . . . . . 13 (𝐸:dom 𝐸1-1-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})))
2516, 17, 243syl 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})))
2625imp 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}))
2726adantl 481 . . . . . . . . . 10 ((((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) ∧ ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉))) ∧ (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}))
2812, 13constr3lem5 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})))
30293ad2ant1 1075 . . . . . . . . . . . . 13 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → (𝐸‘(𝐹‘0)) = (𝐸‘(𝐸‘{𝐴, 𝐵})))
3130eqeq1d 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})))
33323ad2ant2 1076 . . . . . . . . . . . . 13 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → (𝐸‘(𝐹‘1)) = (𝐸‘(𝐸‘{𝐵, 𝐶})))
3433eqeq1d 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})))
36353ad2ant3 1077 . . . . . . . . . . . . 13 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → (𝐸‘(𝐹‘2)) = (𝐸‘(𝐸‘{𝐶, 𝐴})))
3736eqeq1d 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}))
3831, 34, 373anbi123d 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})))
3928, 38ax-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}))
4027, 39sylibr 223 . . . . . . . . 9 ((((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) ∧ ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉))) ∧ (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)
4341, 42preq12d 4220 . . . . . . . . . . . 12 ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → {(𝑃‘0), (𝑃‘1)} = {𝐴, 𝐵})
4443eqeq2d 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)
4642, 45preq12d 4220 . . . . . . . . . . . 12 ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → {(𝑃‘1), (𝑃‘2)} = {𝐵, 𝐶})
4746eqeq2d 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)
4945, 48preq12d 4220 . . . . . . . . . . . 12 ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → {(𝑃‘2), (𝑃‘3)} = {𝐶, 𝐴})
5049eqeq2d 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}))
5144, 47, 503anbi123d 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})))
5251ad2antrr 758 . . . . . . . . 9 ((((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) ∧ ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉))) ∧ (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})))
5340, 52mpbird 246 . . . . . . . 8 ((((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) ∧ ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉))) ∧ (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)}))
5453ex 449 . . . . . . 7 (((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) ∧ ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉))) → ((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)})))
5554ex 449 . . . . . 6 ((((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)) → (((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)) → ((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)}))))
5615, 55mpcom 37 . . . . 5 (((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)) → ((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)})))
57563adant3 1074 . . . 4 (((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ((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)})))
588, 57mpcom 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 → (𝐹𝑘) = (𝐹‘0))
6362fveq2d 6107 . . . . . 6 (𝑘 = 0 → (𝐸‘(𝐹𝑘)) = (𝐸‘(𝐹‘0)))
64 fveq2 6103 . . . . . . 7 (𝑘 = 0 → (𝑃𝑘) = (𝑃‘0))
65 oveq1 6556 . . . . . . . . 9 (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
66 0p1e1 11009 . . . . . . . . 9 (0 + 1) = 1
6765, 66syl6eq 2660 . . . . . . . 8 (𝑘 = 0 → (𝑘 + 1) = 1)
6867fveq2d 6107 . . . . . . 7 (𝑘 = 0 → (𝑃‘(𝑘 + 1)) = (𝑃‘1))
6964, 68preq12d 4220 . . . . . 6 (𝑘 = 0 → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘0), (𝑃‘1)})
7063, 69eqeq12d 2625 . . . . 5 (𝑘 = 0 → ((𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
71 fveq2 6103 . . . . . . 7 (𝑘 = 1 → (𝐹𝑘) = (𝐹‘1))
7271fveq2d 6107 . . . . . 6 (𝑘 = 1 → (𝐸‘(𝐹𝑘)) = (𝐸‘(𝐹‘1)))
73 fveq2 6103 . . . . . . 7 (𝑘 = 1 → (𝑃𝑘) = (𝑃‘1))
74 oveq1 6556 . . . . . . . . 9 (𝑘 = 1 → (𝑘 + 1) = (1 + 1))
75 1p1e2 11011 . . . . . . . . 9 (1 + 1) = 2
7674, 75syl6eq 2660 . . . . . . . 8 (𝑘 = 1 → (𝑘 + 1) = 2)
7776fveq2d 6107 . . . . . . 7 (𝑘 = 1 → (𝑃‘(𝑘 + 1)) = (𝑃‘2))
7873, 77preq12d 4220 . . . . . 6 (𝑘 = 1 → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘1), (𝑃‘2)})
7972, 78eqeq12d 2625 . . . . 5 (𝑘 = 1 → ((𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
80 fveq2 6103 . . . . . . 7 (𝑘 = 2 → (𝐹𝑘) = (𝐹‘2))
8180fveq2d 6107 . . . . . 6 (𝑘 = 2 → (𝐸‘(𝐹𝑘)) = (𝐸‘(𝐹‘2)))
82 fveq2 6103 . . . . . . 7 (𝑘 = 2 → (𝑃𝑘) = (𝑃‘2))
83 oveq1 6556 . . . . . . . . 9 (𝑘 = 2 → (𝑘 + 1) = (2 + 1))
84 2p1e3 11028 . . . . . . . . 9 (2 + 1) = 3
8583, 84syl6eq 2660 . . . . . . . 8 (𝑘 = 2 → (𝑘 + 1) = 3)
8685fveq2d 6107 . . . . . . 7 (𝑘 = 2 → (𝑃‘(𝑘 + 1)) = (𝑃‘3))
8782, 86preq12d 4220 . . . . . 6 (𝑘 = 2 → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘2), (𝑃‘3)})
8881, 87eqeq12d 2625 . . . . 5 (𝑘 = 2 → ((𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}))
8970, 79, 88raltpg 4183 . . . 4 ((0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 2 ∈ ℤ) → (∀𝑘 ∈ {0, 1, 2} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)})))
9059, 60, 61, 89mp3an 1416 . . 3 (∀𝑘 ∈ {0, 1, 2} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}))
9158, 90sylibr 223 . 2 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∀𝑘 ∈ {0, 1, 2} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
9212, 13constr3lem2 26174 . . . . . 6 (#‘𝐹) = 3
9392oveq2i 6560 . . . . 5 (0..^(#‘𝐹)) = (0..^3)
94 fzo0to3tp 12421 . . . . 5 (0..^3) = {0, 1, 2}
9593, 94eqtri 2632 . . . 4 (0..^(#‘𝐹)) = {0, 1, 2}
9695a1i 11 . . 3 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (0..^(#‘𝐹)) = {0, 1, 2})
9796raleqdv 3121 . 2 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1, 2} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
9891, 97mpbird 246 1 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
 Colors of variables: wff setvar class 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
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