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Theorem fvf1tp 13746
Description: Values of a one-to-one function between two sets with three elements. Actually, such a function is a bijection. (Contributed by AV, 23-Jul-2025.)
Assertion
Ref Expression
fvf1tp (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))
Proof of Theorem fvf1tp
StepHypRef Expression
1 f1f 6730 . . 3 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → 𝐹:(0..^3)⟶{𝑋, 𝑌, 𝑍})
2 3nn 12258 . . . . 5 3 ∈ ℕ
3 lbfzo0 13652 . . . . 5 (0 ∈ (0..^3) ↔ 3 ∈ ℕ)
42, 3mpbir 232 . . . 4 0 ∈ (0..^3)
54a1i 11 . . 3 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → 0 ∈ (0..^3))
61, 5ffvelcdmd 7033 . 2 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → (𝐹‘0) ∈ {𝑋, 𝑌, 𝑍})
7 1nn0 12451 . . . . 5 1 ∈ ℕ0
8 1lt3 12347 . . . . 5 1 < 3
9 elfzo0 13653 . . . . 5 (1 ∈ (0..^3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ ∧ 1 < 3))
107, 2, 8, 9mpbir3an 1348 . . . 4 1 ∈ (0..^3)
1110a1i 11 . . 3 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → 1 ∈ (0..^3))
121, 11ffvelcdmd 7033 . 2 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → (𝐹‘1) ∈ {𝑋, 𝑌, 𝑍})
13 2nn0 12452 . . . . 5 2 ∈ ℕ0
14 2lt3 12346 . . . . 5 2 < 3
15 elfzo0 13653 . . . . 5 (2 ∈ (0..^3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ ∧ 2 < 3))
1613, 2, 14, 15mpbir3an 1348 . . . 4 2 ∈ (0..^3)
1716a1i 11 . . 3 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → 2 ∈ (0..^3))
181, 17ffvelcdmd 7033 . 2 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → (𝐹‘2) ∈ {𝑋, 𝑌, 𝑍})
19 eltpi 4627 . . . 4 ((𝐹‘0) ∈ {𝑋, 𝑌, 𝑍} → ((𝐹‘0) = 𝑋 ∨ (𝐹‘0) = 𝑌 ∨ (𝐹‘0) = 𝑍))
20 eltpi 4627 . . . 4 ((𝐹‘1) ∈ {𝑋, 𝑌, 𝑍} → ((𝐹‘1) = 𝑋 ∨ (𝐹‘1) = 𝑌 ∨ (𝐹‘1) = 𝑍))
21 eltpi 4627 . . . 4 ((𝐹‘2) ∈ {𝑋, 𝑌, 𝑍} → ((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍))
2219, 20, 213anim123i 1157 . . 3 (((𝐹‘0) ∈ {𝑋, 𝑌, 𝑍} ∧ (𝐹‘1) ∈ {𝑋, 𝑌, 𝑍} ∧ (𝐹‘2) ∈ {𝑋, 𝑌, 𝑍}) → (((𝐹‘0) = 𝑋 ∨ (𝐹‘0) = 𝑌 ∨ (𝐹‘0) = 𝑍) ∧ ((𝐹‘1) = 𝑋 ∨ (𝐹‘1) = 𝑌 ∨ (𝐹‘1) = 𝑍) ∧ ((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍)))
23 eqeq2 2752 . . . . . . . . . 10 (𝑋 = (𝐹‘0) → ((𝐹‘1) = 𝑋 ↔ (𝐹‘1) = (𝐹‘0)))
2423eqcoms 2748 . . . . . . . . 9 ((𝐹‘0) = 𝑋 → ((𝐹‘1) = 𝑋 ↔ (𝐹‘1) = (𝐹‘0)))
2524adantl 482 . . . . . . . 8 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘1) = 𝑋 ↔ (𝐹‘1) = (𝐹‘0)))
26 f1veqaeq 7207 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (1 ∈ (0..^3) ∧ 0 ∈ (0..^3))) → ((𝐹‘1) = (𝐹‘0) → 1 = 0))
2710, 4, 26mpanr12 711 . . . . . . . . . 10 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((𝐹‘1) = (𝐹‘0) → 1 = 0))
28 ax-1ne0 11105 . . . . . . . . . 10 1 ≠ 0
29 eqneqall 2946 . . . . . . . . . 10 (1 = 0 → (1 ≠ 0 → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
3027, 28, 29syl6mpi 67 . . . . . . . . 9 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((𝐹‘1) = (𝐹‘0) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
3130adantr 481 . . . . . . . 8 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘1) = (𝐹‘0) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
3225, 31sylbid 241 . . . . . . 7 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘1) = 𝑋 → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
33 eqeq2 2752 . . . . . . . . . . . . 13 (𝑋 = (𝐹‘0) → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘0)))
3433eqcoms 2748 . . . . . . . . . . . 12 ((𝐹‘0) = 𝑋 → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘0)))
3534adantl 482 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘0)))
3616, 4pm3.2i 471 . . . . . . . . . . . . . 14 (2 ∈ (0..^3) ∧ 0 ∈ (0..^3))
3736a1i 11 . . . . . . . . . . . . 13 ((𝐹‘0) = 𝑋 → (2 ∈ (0..^3) ∧ 0 ∈ (0..^3)))
38 f1veqaeq 7207 . . . . . . . . . . . . 13 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (2 ∈ (0..^3) ∧ 0 ∈ (0..^3))) → ((𝐹‘2) = (𝐹‘0) → 2 = 0))
3937, 38sylan2 599 . . . . . . . . . . . 12 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘2) = (𝐹‘0) → 2 = 0))
40 2ne0 12283 . . . . . . . . . . . 12 2 ≠ 0
41 eqneqall 2946 . . . . . . . . . . . 12 (2 = 0 → (2 ≠ 0 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
4239, 40, 41syl6mpi 67 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘2) = (𝐹‘0) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
4335, 42sylbid 241 . . . . . . . . . 10 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘2) = 𝑋 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
4443adantr 481 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) → ((𝐹‘2) = 𝑋 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
45 eqeq2 2752 . . . . . . . . . . . 12 (𝑌 = (𝐹‘1) → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘1)))
4645eqcoms 2748 . . . . . . . . . . 11 ((𝐹‘1) = 𝑌 → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘1)))
4746adantl 482 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘1)))
4816, 10pm3.2i 471 . . . . . . . . . . . . . 14 (2 ∈ (0..^3) ∧ 1 ∈ (0..^3))
4948a1i 11 . . . . . . . . . . . . 13 ((𝐹‘0) = 𝑋 → (2 ∈ (0..^3) ∧ 1 ∈ (0..^3)))
50 f1veqaeq 7207 . . . . . . . . . . . . 13 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (2 ∈ (0..^3) ∧ 1 ∈ (0..^3))) → ((𝐹‘2) = (𝐹‘1) → 2 = 1))
5149, 50sylan2 599 . . . . . . . . . . . 12 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘2) = (𝐹‘1) → 2 = 1))
52 1ne2 12382 . . . . . . . . . . . . 13 1 ≠ 2
5352necomi 2989 . . . . . . . . . . . 12 2 ≠ 1
54 eqneqall 2946 . . . . . . . . . . . 12 (2 = 1 → (2 ≠ 1 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
5551, 53, 54syl6mpi 67 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘2) = (𝐹‘1) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
5655adantr 481 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) → ((𝐹‘2) = (𝐹‘1) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
5747, 56sylbid 241 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) → ((𝐹‘2) = 𝑌 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
58 simpllr 781 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑍) → (𝐹‘0) = 𝑋)
59 simplr 774 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑍) → (𝐹‘1) = 𝑌)
60 simpr 485 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑍) → (𝐹‘2) = 𝑍)
6158, 59, 603jca 1134 . . . . . . . . . . . 12 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑍) → ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍))
6261orcd 879 . . . . . . . . . . 11 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑍) → (((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)))
63623mix1d 1343 . . . . . . . . . 10 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))
6463ex 413 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) → ((𝐹‘2) = 𝑍 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
6544, 57, 643jaod 1437 . . . . . . . 8 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
6665ex 413 . . . . . . 7 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘1) = 𝑌 → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
6743adantr 481 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) → ((𝐹‘2) = 𝑋 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
68 simpllr 781 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑌) → (𝐹‘0) = 𝑋)
69 simplr 774 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑌) → (𝐹‘1) = 𝑍)
70 simpr 485 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑌) → (𝐹‘2) = 𝑌)
7168, 69, 703jca 1134 . . . . . . . . . . . 12 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑌) → ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌))
7271olcd 880 . . . . . . . . . . 11 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑌) → (((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)))
73723mix1d 1343 . . . . . . . . . 10 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑌) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))
7473ex 413 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) → ((𝐹‘2) = 𝑌 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
75 eqeq2 2752 . . . . . . . . . . . 12 (𝑍 = (𝐹‘1) → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘1)))
7675eqcoms 2748 . . . . . . . . . . 11 ((𝐹‘1) = 𝑍 → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘1)))
7776adantl 482 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘1)))
7816, 10, 50mpanr12 711 . . . . . . . . . . . . 13 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((𝐹‘2) = (𝐹‘1) → 2 = 1))
7978, 53, 54syl6mpi 67 . . . . . . . . . . . 12 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((𝐹‘2) = (𝐹‘1) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
8079adantr 481 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘2) = (𝐹‘1) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
8180adantr 481 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) → ((𝐹‘2) = (𝐹‘1) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
8277, 81sylbid 241 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) → ((𝐹‘2) = 𝑍 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
8367, 74, 823jaod 1437 . . . . . . . 8 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
8483ex 413 . . . . . . 7 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘1) = 𝑍 → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
8532, 66, 843jaod 1437 . . . . . 6 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → (((𝐹‘1) = 𝑋 ∨ (𝐹‘1) = 𝑌 ∨ (𝐹‘1) = 𝑍) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
8685ex 413 . . . . 5 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((𝐹‘0) = 𝑋 → (((𝐹‘1) = 𝑋 ∨ (𝐹‘1) = 𝑌 ∨ (𝐹‘1) = 𝑍) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))))
87 eqeq2 2752 . . . . . . . . . . . 12 (𝑋 = (𝐹‘1) → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘1)))
8887eqcoms 2748 . . . . . . . . . . 11 ((𝐹‘1) = 𝑋 → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘1)))
8988adantl 482 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘1)))
9079adantr 481 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) → ((𝐹‘2) = (𝐹‘1) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
9190adantr 481 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) → ((𝐹‘2) = (𝐹‘1) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
9289, 91sylbid 241 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) → ((𝐹‘2) = 𝑋 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
93 eqeq2 2752 . . . . . . . . . . . . 13 (𝑌 = (𝐹‘0) → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘0)))
9493eqcoms 2748 . . . . . . . . . . . 12 ((𝐹‘0) = 𝑌 → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘0)))
9594adantl 482 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘0)))
9616, 4, 38mpanr12 711 . . . . . . . . . . . . 13 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((𝐹‘2) = (𝐹‘0) → 2 = 0))
9796, 40, 41syl6mpi 67 . . . . . . . . . . . 12 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((𝐹‘2) = (𝐹‘0) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
9897adantr 481 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) → ((𝐹‘2) = (𝐹‘0) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
9995, 98sylbid 241 . . . . . . . . . 10 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) → ((𝐹‘2) = 𝑌 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
10099adantr 481 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) → ((𝐹‘2) = 𝑌 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
101 simpllr 781 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑍) → (𝐹‘0) = 𝑌)
102 simplr 774 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑍) → (𝐹‘1) = 𝑋)
103 simpr 485 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑍) → (𝐹‘2) = 𝑍)
104101, 102, 1033jca 1134 . . . . . . . . . . . 12 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑍) → ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍))
105104orcd 879 . . . . . . . . . . 11 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑍) → (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)))
1061053mix2d 1344 . . . . . . . . . 10 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))
107106ex 413 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) → ((𝐹‘2) = 𝑍 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
10892, 100, 1073jaod 1437 . . . . . . . 8 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
109108ex 413 . . . . . . 7 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) → ((𝐹‘1) = 𝑋 → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
110 eqeq2 2752 . . . . . . . . . 10 (𝑌 = (𝐹‘0) → ((𝐹‘1) = 𝑌 ↔ (𝐹‘1) = (𝐹‘0)))
111110eqcoms 2748 . . . . . . . . 9 ((𝐹‘0) = 𝑌 → ((𝐹‘1) = 𝑌 ↔ (𝐹‘1) = (𝐹‘0)))
112111adantl 482 . . . . . . . 8 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) → ((𝐹‘1) = 𝑌 ↔ (𝐹‘1) = (𝐹‘0)))
11330adantr 481 . . . . . . . 8 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) → ((𝐹‘1) = (𝐹‘0) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
114112, 113sylbid 241 . . . . . . 7 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) → ((𝐹‘1) = 𝑌 → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
115 simpllr 781 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑋) → (𝐹‘0) = 𝑌)
116 simplr 774 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑋) → (𝐹‘1) = 𝑍)
117 simpr 485 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑋) → (𝐹‘2) = 𝑋)
118115, 116, 1173jca 1134 . . . . . . . . . . . 12 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑋) → ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋))
119118olcd 880 . . . . . . . . . . 11 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑋) → (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)))
1201193mix2d 1344 . . . . . . . . . 10 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑋) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))
121120ex 413 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) → ((𝐹‘2) = 𝑋 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
12299adantr 481 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) → ((𝐹‘2) = 𝑌 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
12376adantl 482 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘1)))
12490adantr 481 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) → ((𝐹‘2) = (𝐹‘1) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
125123, 124sylbid 241 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) → ((𝐹‘2) = 𝑍 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
126121, 122, 1253jaod 1437 . . . . . . . 8 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
127126ex 413 . . . . . . 7 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) → ((𝐹‘1) = 𝑍 → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
128109, 114, 1273jaod 1437 . . . . . 6 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) → (((𝐹‘1) = 𝑋 ∨ (𝐹‘1) = 𝑌 ∨ (𝐹‘1) = 𝑍) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
129128ex 413 . . . . 5 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((𝐹‘0) = 𝑌 → (((𝐹‘1) = 𝑋 ∨ (𝐹‘1) = 𝑌 ∨ (𝐹‘1) = 𝑍) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))))
13088adantl 482 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘1) = 𝑋) → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘1)))
13179adantr 481 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘1) = 𝑋) → ((𝐹‘2) = (𝐹‘1) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
132130, 131sylbid 241 . . . . . . . . . 10 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘1) = 𝑋) → ((𝐹‘2) = 𝑋 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
133132adantlr 721 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) → ((𝐹‘2) = 𝑋 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
134 simpllr 781 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑌) → (𝐹‘0) = 𝑍)
135 simplr 774 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑌) → (𝐹‘1) = 𝑋)
136 simpr 485 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑌) → (𝐹‘2) = 𝑌)
137134, 135, 1363jca 1134 . . . . . . . . . . . 12 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑌) → ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌))
138137orcd 879 . . . . . . . . . . 11 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑌) → (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))
1391383mix3d 1345 . . . . . . . . . 10 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑌) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))
140139ex 413 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) → ((𝐹‘2) = 𝑌 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
141 eqeq2 2752 . . . . . . . . . . . . 13 (𝑍 = (𝐹‘0) → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘0)))
142141eqcoms 2748 . . . . . . . . . . . 12 ((𝐹‘0) = 𝑍 → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘0)))
143142adantl 482 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘0)))
14497adantr 481 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) → ((𝐹‘2) = (𝐹‘0) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
145143, 144sylbid 241 . . . . . . . . . 10 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) → ((𝐹‘2) = 𝑍 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
146145adantr 481 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) → ((𝐹‘2) = 𝑍 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
147133, 140, 1463jaod 1437 . . . . . . . 8 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
148147ex 413 . . . . . . 7 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) → ((𝐹‘1) = 𝑋 → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
149 simpllr 781 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑋) → (𝐹‘0) = 𝑍)
150 simplr 774 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑋) → (𝐹‘1) = 𝑌)
151 simpr 485 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑋) → (𝐹‘2) = 𝑋)
152149, 150, 1513jca 1134 . . . . . . . . . . . 12 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑋) → ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))
153152olcd 880 . . . . . . . . . . 11 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑋) → (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))
1541533mix3d 1345 . . . . . . . . . 10 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑋) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))
155154ex 413 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) → ((𝐹‘2) = 𝑋 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
15646adantl 482 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘1)))
15779adantr 481 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) → ((𝐹‘2) = (𝐹‘1) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
158157adantr 481 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) → ((𝐹‘2) = (𝐹‘1) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
159156, 158sylbid 241 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) → ((𝐹‘2) = 𝑌 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
160145adantr 481 . . . . . . . . 9 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) → ((𝐹‘2) = 𝑍 → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
161155, 159, 1603jaod 1437 . . . . . . . 8 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
162161ex 413 . . . . . . 7 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) → ((𝐹‘1) = 𝑌 → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
163 eqeq2 2752 . . . . . . . . . 10 (𝑍 = (𝐹‘0) → ((𝐹‘1) = 𝑍 ↔ (𝐹‘1) = (𝐹‘0)))
164163eqcoms 2748 . . . . . . . . 9 ((𝐹‘0) = 𝑍 → ((𝐹‘1) = 𝑍 ↔ (𝐹‘1) = (𝐹‘0)))
165164adantl 482 . . . . . . . 8 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) → ((𝐹‘1) = 𝑍 ↔ (𝐹‘1) = (𝐹‘0)))
16630adantr 481 . . . . . . . 8 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) → ((𝐹‘1) = (𝐹‘0) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
167165, 166sylbid 241 . . . . . . 7 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) → ((𝐹‘1) = 𝑍 → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
168148, 162, 1673jaod 1437 . . . . . 6 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) → (((𝐹‘1) = 𝑋 ∨ (𝐹‘1) = 𝑌 ∨ (𝐹‘1) = 𝑍) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))))
169168ex 413 . . . . 5 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((𝐹‘0) = 𝑍 → (((𝐹‘1) = 𝑋 ∨ (𝐹‘1) = 𝑌 ∨ (𝐹‘1) = 𝑍) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))))
17086, 129, 1693jaod 1437 . . . 4 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → (((𝐹‘0) = 𝑋 ∨ (𝐹‘0) = 𝑌 ∨ (𝐹‘0) = 𝑍) → (((𝐹‘1) = 𝑋 ∨ (𝐹‘1) = 𝑌 ∨ (𝐹‘1) = 𝑍) → (((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))))
1711703impd 1355 . . 3 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((((𝐹‘0) = 𝑋 ∨ (𝐹‘0) = 𝑌 ∨ (𝐹‘0) = 𝑍) ∧ ((𝐹‘1) = 𝑋 ∨ (𝐹‘1) = 𝑌 ∨ (𝐹‘1) = 𝑍) ∧ ((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍)) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
17222, 171syl5 34 . 2 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → (((𝐹‘0) ∈ {𝑋, 𝑌, 𝑍} ∧ (𝐹‘1) ∈ {𝑋, 𝑌, 𝑍} ∧ (𝐹‘2) ∈ {𝑋, 𝑌, 𝑍}) → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋)))))
1736, 12, 18, 172mp3and 1472 1 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∨ w3o 1091   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  {ctp 4566   class class class wbr 5079  –1-1→wf1 6489  ‘cfv 6492  (class class class)co 7363  0cc0 11036  1c1 11037   < clt 11177  ℕcn 12172  2c2 12234  3c3 12235  ℕ0cn0 12435  ..^cfzo 13606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-n0 12436  df-z 12523  df-uz 12787  df-fz 13460  df-fzo 13607
This theorem is referenced by:  grtriproplem  48431  grtrif1o  48434
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