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Theorem fvf1tp 13898
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 6766 . . 3 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → 𝐹:(0..^3)⟶{𝑋, 𝑌, 𝑍})
2 3nn 12392 . . . . 5 3 ∈ ℕ
3 lbfzo0 13803 . . . . 5 (0 ∈ (0..^3) ↔ 3 ∈ ℕ)
42, 3mpbir 234 . . . 4 0 ∈ (0..^3)
54a1i 11 . . 3 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → 0 ∈ (0..^3))
61, 5ffvelcdmd 7073 . 2 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → (𝐹‘0) ∈ {𝑋, 𝑌, 𝑍})
7 1nn0 12592 . . . . 5 1 ∈ ℕ0
8 1lt3 12488 . . . . 5 1 < 3
9 elfzo0 13804 . . . . 5 (1 ∈ (0..^3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ ∧ 1 < 3))
107, 2, 8, 9mpbir3an 1360 . . . 4 1 ∈ (0..^3)
1110a1i 11 . . 3 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → 1 ∈ (0..^3))
121, 11ffvelcdmd 7073 . 2 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → (𝐹‘1) ∈ {𝑋, 𝑌, 𝑍})
13 2nn0 12593 . . . . 5 2 ∈ ℕ0
14 2lt3 12486 . . . . 5 2 < 3
15 elfzo0 13804 . . . . 5 (2 ∈ (0..^3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ ∧ 2 < 3))
1613, 2, 14, 15mpbir3an 1360 . . . 4 2 ∈ (0..^3)
1716a1i 11 . . 3 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → 2 ∈ (0..^3))
181, 17ffvelcdmd 7073 . 2 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → (𝐹‘2) ∈ {𝑋, 𝑌, 𝑍})
19 eltpi 4648 . . . 4 ((𝐹‘0) ∈ {𝑋, 𝑌, 𝑍} → ((𝐹‘0) = 𝑋 ∨ (𝐹‘0) = 𝑌 ∨ (𝐹‘0) = 𝑍))
20 eltpi 4648 . . . 4 ((𝐹‘1) ∈ {𝑋, 𝑌, 𝑍} → ((𝐹‘1) = 𝑋 ∨ (𝐹‘1) = 𝑌 ∨ (𝐹‘1) = 𝑍))
21 eltpi 4648 . . . 4 ((𝐹‘2) ∈ {𝑋, 𝑌, 𝑍} → ((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍))
2219, 20, 213anim123i 1169 . . 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 2772 . . . . . . . . . 10 (𝑋 = (𝐹‘0) → ((𝐹‘1) = 𝑋 ↔ (𝐹‘1) = (𝐹‘0)))
2423eqcoms 2768 . . . . . . . . 9 ((𝐹‘0) = 𝑋 → ((𝐹‘1) = 𝑋 ↔ (𝐹‘1) = (𝐹‘0)))
2524adantl 487 . . . . . . . 8 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘1) = 𝑋 ↔ (𝐹‘1) = (𝐹‘0)))
26 f1veqaeq 7248 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (1 ∈ (0..^3) ∧ 0 ∈ (0..^3))) → ((𝐹‘1) = (𝐹‘0) → 1 = 0))
2710, 4, 26mpanr12 718 . . . . . . . . . 10 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((𝐹‘1) = (𝐹‘0) → 1 = 0))
28 ax-1ne0 11241 . . . . . . . . . 10 1 ≠ 0
29 eqneqall 2966 . . . . . . . . . 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 68 . . . . . . . . 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 486 . . . . . . . 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 243 . . . . . . 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 2772 . . . . . . . . . . . . 13 (𝑋 = (𝐹‘0) → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘0)))
3433eqcoms 2768 . . . . . . . . . . . 12 ((𝐹‘0) = 𝑋 → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘0)))
3534adantl 487 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘0)))
3616, 4pm3.2i 476 . . . . . . . . . . . . . 14 (2 ∈ (0..^3) ∧ 0 ∈ (0..^3))
3736a1i 11 . . . . . . . . . . . . 13 ((𝐹‘0) = 𝑋 → (2 ∈ (0..^3) ∧ 0 ∈ (0..^3)))
38 f1veqaeq 7248 . . . . . . . . . . . . 13 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (2 ∈ (0..^3) ∧ 0 ∈ (0..^3))) → ((𝐹‘2) = (𝐹‘0) → 2 = 0))
3937, 38sylan2 605 . . . . . . . . . . . 12 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘2) = (𝐹‘0) → 2 = 0))
40 2ne0 12419 . . . . . . . . . . . 12 2 ≠ 0
41 eqneqall 2966 . . . . . . . . . . . 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 68 . . . . . . . . . . 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 243 . . . . . . . . . 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 486 . . . . . . . . 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 2772 . . . . . . . . . . . 12 (𝑌 = (𝐹‘1) → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘1)))
4645eqcoms 2768 . . . . . . . . . . 11 ((𝐹‘1) = 𝑌 → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘1)))
4746adantl 487 . . . . . . . . . 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 476 . . . . . . . . . . . . . 14 (2 ∈ (0..^3) ∧ 1 ∈ (0..^3))
4948a1i 11 . . . . . . . . . . . . 13 ((𝐹‘0) = 𝑋 → (2 ∈ (0..^3) ∧ 1 ∈ (0..^3)))
50 f1veqaeq 7248 . . . . . . . . . . . . 13 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (2 ∈ (0..^3) ∧ 1 ∈ (0..^3))) → ((𝐹‘2) = (𝐹‘1) → 2 = 1))
5149, 50sylan2 605 . . . . . . . . . . . 12 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘2) = (𝐹‘1) → 2 = 1))
52 1ne2 12523 . . . . . . . . . . . . 13 1 ≠ 2
5352necomi 3009 . . . . . . . . . . . 12 2 ≠ 1
54 eqneqall 2966 . . . . . . . . . . . 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 68 . . . . . . . . . . 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 486 . . . . . . . . . 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 243 . . . . . . . . 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 788 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑍) → (𝐹‘0) = 𝑋)
59 simplr 781 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑍) → (𝐹‘1) = 𝑌)
60 simpr 490 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑍) → (𝐹‘2) = 𝑍)
6158, 59, 603jca 1146 . . . . . . . . . . . 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 887 . . . . . . . . . . 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 1355 . . . . . . . . . 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 418 . . . . . . . . 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 1456 . . . . . . . 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 418 . . . . . . 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 486 . . . . . . . . 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 788 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑌) → (𝐹‘0) = 𝑋)
69 simplr 781 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑌) → (𝐹‘1) = 𝑍)
70 simpr 490 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑌) → (𝐹‘2) = 𝑌)
7168, 69, 703jca 1146 . . . . . . . . . . . 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 888 . . . . . . . . . . 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 1355 . . . . . . . . . 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 418 . . . . . . . . 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 2772 . . . . . . . . . . . 12 (𝑍 = (𝐹‘1) → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘1)))
7675eqcoms 2768 . . . . . . . . . . 11 ((𝐹‘1) = 𝑍 → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘1)))
7776adantl 487 . . . . . . . . . 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 718 . . . . . . . . . . . . 13 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((𝐹‘2) = (𝐹‘1) → 2 = 1))
7978, 53, 54syl6mpi 68 . . . . . . . . . . . 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 486 . . . . . . . . . . 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 486 . . . . . . . . . 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 243 . . . . . . . . 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 1456 . . . . . . . 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 418 . . . . . . 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 1456 . . . . . 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 418 . . . . 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 2772 . . . . . . . . . . . 12 (𝑋 = (𝐹‘1) → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘1)))
8887eqcoms 2768 . . . . . . . . . . 11 ((𝐹‘1) = 𝑋 → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘1)))
8988adantl 487 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘1)))
9079adantr 486 . . . . . . . . . . 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 486 . . . . . . . . . 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 243 . . . . . . . . 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 2772 . . . . . . . . . . . . 13 (𝑌 = (𝐹‘0) → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘0)))
9493eqcoms 2768 . . . . . . . . . . . 12 ((𝐹‘0) = 𝑌 → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘0)))
9594adantl 487 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘0)))
9616, 4, 38mpanr12 718 . . . . . . . . . . . . 13 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((𝐹‘2) = (𝐹‘0) → 2 = 0))
9796, 40, 41syl6mpi 68 . . . . . . . . . . . 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 486 . . . . . . . . . . 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 243 . . . . . . . . . 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 486 . . . . . . . . 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 788 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑍) → (𝐹‘0) = 𝑌)
102 simplr 781 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑍) → (𝐹‘1) = 𝑋)
103 simpr 490 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑍) → (𝐹‘2) = 𝑍)
104101, 102, 1033jca 1146 . . . . . . . . . . . 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 887 . . . . . . . . . . 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 1356 . . . . . . . . . 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 418 . . . . . . . . 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 1456 . . . . . . . 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 418 . . . . . . 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 2772 . . . . . . . . . 10 (𝑌 = (𝐹‘0) → ((𝐹‘1) = 𝑌 ↔ (𝐹‘1) = (𝐹‘0)))
111110eqcoms 2768 . . . . . . . . 9 ((𝐹‘0) = 𝑌 → ((𝐹‘1) = 𝑌 ↔ (𝐹‘1) = (𝐹‘0)))
112111adantl 487 . . . . . . . 8 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) → ((𝐹‘1) = 𝑌 ↔ (𝐹‘1) = (𝐹‘0)))
11330adantr 486 . . . . . . . 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 243 . . . . . . 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 788 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑋) → (𝐹‘0) = 𝑌)
116 simplr 781 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑋) → (𝐹‘1) = 𝑍)
117 simpr 490 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑋) → (𝐹‘2) = 𝑋)
118115, 116, 1173jca 1146 . . . . . . . . . . . 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 888 . . . . . . . . . . 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 1356 . . . . . . . . . 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 418 . . . . . . . . 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 486 . . . . . . . . 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 487 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘1)))
12490adantr 486 . . . . . . . . . 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 243 . . . . . . . . 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 1456 . . . . . . . 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 418 . . . . . . 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 1456 . . . . . 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 418 . . . . 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 487 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘1) = 𝑋) → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘1)))
13179adantr 486 . . . . . . . . . . 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 243 . . . . . . . . . 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 728 . . . . . . . . 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 788 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑌) → (𝐹‘0) = 𝑍)
135 simplr 781 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑌) → (𝐹‘1) = 𝑋)
136 simpr 490 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑌) → (𝐹‘2) = 𝑌)
137134, 135, 1363jca 1146 . . . . . . . . . . . 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 887 . . . . . . . . . . 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 1357 . . . . . . . . . 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 418 . . . . . . . . 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 2772 . . . . . . . . . . . . 13 (𝑍 = (𝐹‘0) → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘0)))
142141eqcoms 2768 . . . . . . . . . . . 12 ((𝐹‘0) = 𝑍 → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘0)))
143142adantl 487 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘0)))
14497adantr 486 . . . . . . . . . . 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 243 . . . . . . . . . 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 486 . . . . . . . . 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 1456 . . . . . . . 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 418 . . . . . . 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 788 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑋) → (𝐹‘0) = 𝑍)
150 simplr 781 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑋) → (𝐹‘1) = 𝑌)
151 simpr 490 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑋) → (𝐹‘2) = 𝑋)
152149, 150, 1513jca 1146 . . . . . . . . . . . 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 888 . . . . . . . . . . 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 1357 . . . . . . . . . 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 418 . . . . . . . . 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 487 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘1)))
15779adantr 486 . . . . . . . . . . 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 486 . . . . . . . . . 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 243 . . . . . . . . 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 486 . . . . . . . . 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 1456 . . . . . . . 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 418 . . . . . . 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 2772 . . . . . . . . . 10 (𝑍 = (𝐹‘0) → ((𝐹‘1) = 𝑍 ↔ (𝐹‘1) = (𝐹‘0)))
164163eqcoms 2768 . . . . . . . . 9 ((𝐹‘0) = 𝑍 → ((𝐹‘1) = 𝑍 ↔ (𝐹‘1) = (𝐹‘0)))
165164adantl 487 . . . . . . . 8 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) → ((𝐹‘1) = 𝑍 ↔ (𝐹‘1) = (𝐹‘0)))
16630adantr 486 . . . . . . . 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 243 . . . . . . 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 1456 . . . . . 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 418 . . . . 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 1456 . . . 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 1367 . . 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 35 . 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 1493 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
This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  {ctp 4587   class class class wbr 5102  –1-1→wf1 6524  ‘cfv 6527  (class class class)co 7408  0cc0 11172  1c1 11173   < clt 11315  ℕcn 12305  2c2 12367  3c3 12368  ℕ0cn0 12576  ..^cfzo 13757
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11228  ax-resscn 11229  ax-1cn 11230  ax-icn 11231  ax-addcl 11232  ax-addrcl 11233  ax-mulcl 11234  ax-mulrcl 11235  ax-mulcom 11236  ax-addass 11237  ax-mulass 11238  ax-distr 11239  ax-i2m1 11240  ax-1ne0 11241  ax-1rid 11242  ax-rnegex 11243  ax-rrecex 11244  ax-cnre 11245  ax-pre-lttri 11246  ax-pre-lttrn 11247  ax-pre-ltadd 11248  ax-pre-mulgt0 11249
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8695  df-en 8952  df-dom 8953  df-sdom 8954  df-pnf 11317  df-mnf 11318  df-xr 11319  df-ltxr 11320  df-le 11321  df-sub 11515  df-neg 11516  df-nn 12306  df-2 12375  df-3 12376  df-n0 12577  df-z 12664  df-uz 12936  df-fz 13610  df-fzo 13758
This theorem is used by:  grtriproplem  48959  grtrif1o  48962
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