MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvf1tp Structured version   Visualization version   GIF version
Theorem fvf1tp 13721
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 6738 . . 3 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → 𝐹:(0..^3)⟶{𝑋, 𝑌, 𝑍})
2 3nn 12236 . . . . 5 3 ∈ ℕ
3 lbfzo0 13627 . . . . 5 (0 ∈ (0..^3) ↔ 3 ∈ ℕ)
42, 3mpbir 231 . . . 4 0 ∈ (0..^3)
54a1i 11 . . 3 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → 0 ∈ (0..^3))
61, 5ffvelcdmd 7039 . 2 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → (𝐹‘0) ∈ {𝑋, 𝑌, 𝑍})
7 1nn0 12429 . . . . 5 1 ∈ ℕ0
8 1lt3 12325 . . . . 5 1 < 3
9 elfzo0 13628 . . . . 5 (1 ∈ (0..^3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ ∧ 1 < 3))
107, 2, 8, 9mpbir3an 1343 . . . 4 1 ∈ (0..^3)
1110a1i 11 . . 3 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → 1 ∈ (0..^3))
121, 11ffvelcdmd 7039 . 2 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → (𝐹‘1) ∈ {𝑋, 𝑌, 𝑍})
13 2nn0 12430 . . . . 5 2 ∈ ℕ0
14 2lt3 12324 . . . . 5 2 < 3
15 elfzo0 13628 . . . . 5 (2 ∈ (0..^3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ ∧ 2 < 3))
1613, 2, 14, 15mpbir3an 1343 . . . 4 2 ∈ (0..^3)
1716a1i 11 . . 3 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → 2 ∈ (0..^3))
181, 17ffvelcdmd 7039 . 2 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → (𝐹‘2) ∈ {𝑋, 𝑌, 𝑍})
19 eltpi 4647 . . . 4 ((𝐹‘0) ∈ {𝑋, 𝑌, 𝑍} → ((𝐹‘0) = 𝑋 ∨ (𝐹‘0) = 𝑌 ∨ (𝐹‘0) = 𝑍))
20 eltpi 4647 . . . 4 ((𝐹‘1) ∈ {𝑋, 𝑌, 𝑍} → ((𝐹‘1) = 𝑋 ∨ (𝐹‘1) = 𝑌 ∨ (𝐹‘1) = 𝑍))
21 eltpi 4647 . . . 4 ((𝐹‘2) ∈ {𝑋, 𝑌, 𝑍} → ((𝐹‘2) = 𝑋 ∨ (𝐹‘2) = 𝑌 ∨ (𝐹‘2) = 𝑍))
2219, 20, 213anim123i 1152 . . 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 2749 . . . . . . . . . 10 (𝑋 = (𝐹‘0) → ((𝐹‘1) = 𝑋 ↔ (𝐹‘1) = (𝐹‘0)))
2423eqcoms 2745 . . . . . . . . 9 ((𝐹‘0) = 𝑋 → ((𝐹‘1) = 𝑋 ↔ (𝐹‘1) = (𝐹‘0)))
2524adantl 481 . . . . . . . 8 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘1) = 𝑋 ↔ (𝐹‘1) = (𝐹‘0)))
26 f1veqaeq 7212 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (1 ∈ (0..^3) ∧ 0 ∈ (0..^3))) → ((𝐹‘1) = (𝐹‘0) → 1 = 0))
2710, 4, 26mpanr12 706 . . . . . . . . . 10 (𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((𝐹‘1) = (𝐹‘0) → 1 = 0))
28 ax-1ne0 11107 . . . . . . . . . 10 1 ≠ 0
29 eqneqall 2944 . . . . . . . . . 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 480 . . . . . . . 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 240 . . . . . . 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 2749 . . . . . . . . . . . . 13 (𝑋 = (𝐹‘0) → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘0)))
3433eqcoms 2745 . . . . . . . . . . . 12 ((𝐹‘0) = 𝑋 → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘0)))
3534adantl 481 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘0)))
3616, 4pm3.2i 470 . . . . . . . . . . . . . 14 (2 ∈ (0..^3) ∧ 0 ∈ (0..^3))
3736a1i 11 . . . . . . . . . . . . 13 ((𝐹‘0) = 𝑋 → (2 ∈ (0..^3) ∧ 0 ∈ (0..^3)))
38 f1veqaeq 7212 . . . . . . . . . . . . 13 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (2 ∈ (0..^3) ∧ 0 ∈ (0..^3))) → ((𝐹‘2) = (𝐹‘0) → 2 = 0))
3937, 38sylan2 594 . . . . . . . . . . . 12 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘2) = (𝐹‘0) → 2 = 0))
40 2ne0 12261 . . . . . . . . . . . 12 2 ≠ 0
41 eqneqall 2944 . . . . . . . . . . . 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 240 . . . . . . . . . 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 480 . . . . . . . . 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 2749 . . . . . . . . . . . 12 (𝑌 = (𝐹‘1) → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘1)))
4645eqcoms 2745 . . . . . . . . . . 11 ((𝐹‘1) = 𝑌 → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘1)))
4746adantl 481 . . . . . . . . . 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 470 . . . . . . . . . . . . . 14 (2 ∈ (0..^3) ∧ 1 ∈ (0..^3))
4948a1i 11 . . . . . . . . . . . . 13 ((𝐹‘0) = 𝑋 → (2 ∈ (0..^3) ∧ 1 ∈ (0..^3)))
50 f1veqaeq 7212 . . . . . . . . . . . . 13 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (2 ∈ (0..^3) ∧ 1 ∈ (0..^3))) → ((𝐹‘2) = (𝐹‘1) → 2 = 1))
5149, 50sylan2 594 . . . . . . . . . . . 12 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) → ((𝐹‘2) = (𝐹‘1) → 2 = 1))
52 1ne2 12360 . . . . . . . . . . . . 13 1 ≠ 2
5352necomi 2987 . . . . . . . . . . . 12 2 ≠ 1
54 eqneqall 2944 . . . . . . . . . . . 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 480 . . . . . . . . . 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 240 . . . . . . . . 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 776 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑍) → (𝐹‘0) = 𝑋)
59 simplr 769 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑍) → (𝐹‘1) = 𝑌)
60 simpr 484 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑍) → (𝐹‘2) = 𝑍)
6158, 59, 603jca 1129 . . . . . . . . . . . 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 874 . . . . . . . . . . 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 1338 . . . . . . . . . 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 412 . . . . . . . . 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 1432 . . . . . . . 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 412 . . . . . . 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 480 . . . . . . . . 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 776 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑌) → (𝐹‘0) = 𝑋)
69 simplr 769 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑌) → (𝐹‘1) = 𝑍)
70 simpr 484 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑋) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑌) → (𝐹‘2) = 𝑌)
7168, 69, 703jca 1129 . . . . . . . . . . . 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 875 . . . . . . . . . . 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 1338 . . . . . . . . . 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 412 . . . . . . . . 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 2749 . . . . . . . . . . . 12 (𝑍 = (𝐹‘1) → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘1)))
7675eqcoms 2745 . . . . . . . . . . 11 ((𝐹‘1) = 𝑍 → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘1)))
7776adantl 481 . . . . . . . . . 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 706 . . . . . . . . . . . . 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 480 . . . . . . . . . . 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 480 . . . . . . . . . 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 240 . . . . . . . . 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 1432 . . . . . . . 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 412 . . . . . . 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 1432 . . . . . 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 412 . . . . 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 2749 . . . . . . . . . . . 12 (𝑋 = (𝐹‘1) → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘1)))
8887eqcoms 2745 . . . . . . . . . . 11 ((𝐹‘1) = 𝑋 → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘1)))
8988adantl 481 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘1)))
9079adantr 480 . . . . . . . . . . 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 480 . . . . . . . . . 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 240 . . . . . . . . 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 2749 . . . . . . . . . . . . 13 (𝑌 = (𝐹‘0) → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘0)))
9493eqcoms 2745 . . . . . . . . . . . 12 ((𝐹‘0) = 𝑌 → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘0)))
9594adantl 481 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘0)))
9616, 4, 38mpanr12 706 . . . . . . . . . . . . 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 480 . . . . . . . . . . 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 240 . . . . . . . . . 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 480 . . . . . . . . 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 776 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑍) → (𝐹‘0) = 𝑌)
102 simplr 769 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑍) → (𝐹‘1) = 𝑋)
103 simpr 484 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑍) → (𝐹‘2) = 𝑍)
104101, 102, 1033jca 1129 . . . . . . . . . . . 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 874 . . . . . . . . . . 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 1339 . . . . . . . . . 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 412 . . . . . . . . 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 1432 . . . . . . . 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 412 . . . . . . 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 2749 . . . . . . . . . 10 (𝑌 = (𝐹‘0) → ((𝐹‘1) = 𝑌 ↔ (𝐹‘1) = (𝐹‘0)))
111110eqcoms 2745 . . . . . . . . 9 ((𝐹‘0) = 𝑌 → ((𝐹‘1) = 𝑌 ↔ (𝐹‘1) = (𝐹‘0)))
112111adantl 481 . . . . . . . 8 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) → ((𝐹‘1) = 𝑌 ↔ (𝐹‘1) = (𝐹‘0)))
11330adantr 480 . . . . . . . 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 240 . . . . . . 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 776 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑋) → (𝐹‘0) = 𝑌)
116 simplr 769 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑋) → (𝐹‘1) = 𝑍)
117 simpr 484 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) ∧ (𝐹‘2) = 𝑋) → (𝐹‘2) = 𝑋)
118115, 116, 1173jca 1129 . . . . . . . . . . . 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 875 . . . . . . . . . . 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 1339 . . . . . . . . . 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 412 . . . . . . . . 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 480 . . . . . . . . 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 481 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑌) ∧ (𝐹‘1) = 𝑍) → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘1)))
12490adantr 480 . . . . . . . . . 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 240 . . . . . . . . 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 1432 . . . . . . . 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 412 . . . . . . 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 1432 . . . . . 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 412 . . . . 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 481 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘1) = 𝑋) → ((𝐹‘2) = 𝑋 ↔ (𝐹‘2) = (𝐹‘1)))
13179adantr 480 . . . . . . . . . . 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 240 . . . . . . . . . 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 716 . . . . . . . . 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 776 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑌) → (𝐹‘0) = 𝑍)
135 simplr 769 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑌) → (𝐹‘1) = 𝑋)
136 simpr 484 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑋) ∧ (𝐹‘2) = 𝑌) → (𝐹‘2) = 𝑌)
137134, 135, 1363jca 1129 . . . . . . . . . . . 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 874 . . . . . . . . . . 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 1340 . . . . . . . . . 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 412 . . . . . . . . 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 2749 . . . . . . . . . . . . 13 (𝑍 = (𝐹‘0) → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘0)))
142141eqcoms 2745 . . . . . . . . . . . 12 ((𝐹‘0) = 𝑍 → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘0)))
143142adantl 481 . . . . . . . . . . 11 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) → ((𝐹‘2) = 𝑍 ↔ (𝐹‘2) = (𝐹‘0)))
14497adantr 480 . . . . . . . . . . 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 240 . . . . . . . . . 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 480 . . . . . . . . 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 1432 . . . . . . . 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 412 . . . . . . 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 776 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑋) → (𝐹‘0) = 𝑍)
150 simplr 769 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑋) → (𝐹‘1) = 𝑌)
151 simpr 484 . . . . . . . . . . . . 13 ((((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) ∧ (𝐹‘2) = 𝑋) → (𝐹‘2) = 𝑋)
152149, 150, 1513jca 1129 . . . . . . . . . . . 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 875 . . . . . . . . . . 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 1340 . . . . . . . . . 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 412 . . . . . . . . 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 481 . . . . . . . . . 10 (((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) ∧ (𝐹‘1) = 𝑌) → ((𝐹‘2) = 𝑌 ↔ (𝐹‘2) = (𝐹‘1)))
15779adantr 480 . . . . . . . . . . 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 480 . . . . . . . . . 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 240 . . . . . . . . 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 480 . . . . . . . . 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 1432 . . . . . . . 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 412 . . . . . . 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 2749 . . . . . . . . . 10 (𝑍 = (𝐹‘0) → ((𝐹‘1) = 𝑍 ↔ (𝐹‘1) = (𝐹‘0)))
164163eqcoms 2745 . . . . . . . . 9 ((𝐹‘0) = 𝑍 → ((𝐹‘1) = 𝑍 ↔ (𝐹‘1) = (𝐹‘0)))
165164adantl 481 . . . . . . . 8 ((𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} ∧ (𝐹‘0) = 𝑍) → ((𝐹‘1) = 𝑍 ↔ (𝐹‘1) = (𝐹‘0)))
16630adantr 480 . . . . . . . 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 240 . . . . . . 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 1432 . . . . . 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 412 . . . . 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 1432 . . . 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 1350 . . 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 1467 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 206   ∧ wa 395   ∨ wo 848   ∨ w3o 1086   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {ctp 4586   class class class wbr 5100  –1-1→wf1 6497  ‘cfv 6500  (class class class)co 7368  0cc0 11038  1c1 11039   < clt 11178  ℕcn 12157  2c2 12212  3c3 12213  ℕ0cn0 12413  ..^cfzo 13582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-z 12501  df-uz 12764  df-fz 13436  df-fzo 13583
This theorem is referenced by:  grtriproplem  48288  grtrif1o  48291
  Copyright terms: Public domain W3C validator