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Theorem injresinj 12629
Description: A function whose restriction is injective and the values of the remaining arguments are different from all other values is injective itself. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
Assertion
Ref Expression
injresinj (𝐾 ∈ ℕ0 → ((𝐹:(0...𝐾)⟶𝑉 ∧ Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹‘0) ≠ (𝐹𝐾)) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun 𝐹)))

Proof of Theorem injresinj
Dummy variables 𝑣 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzo0ss1 12537 . . . . . . . . 9 (1..^𝐾) ⊆ (0..^𝐾)
2 fzossfz 12527 . . . . . . . . 9 (0..^𝐾) ⊆ (0...𝐾)
31, 2sstri 3645 . . . . . . . 8 (1..^𝐾) ⊆ (0...𝐾)
4 fssres 6108 . . . . . . . 8 ((𝐹:(0...𝐾)⟶𝑉 ∧ (1..^𝐾) ⊆ (0...𝐾)) → (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)
53, 4mpan2 707 . . . . . . 7 (𝐹:(0...𝐾)⟶𝑉 → (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)
65biantrud 527 . . . . . 6 (𝐹:(0...𝐾)⟶𝑉 → (Fun (𝐹 ↾ (1..^𝐾)) ↔ (Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)))
7 ancom 465 . . . . . . 7 ((Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉) ↔ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉 ∧ Fun (𝐹 ↾ (1..^𝐾))))
8 df-f1 5931 . . . . . . 7 ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉 ↔ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉 ∧ Fun (𝐹 ↾ (1..^𝐾))))
97, 8bitr4i 267 . . . . . 6 ((Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉) ↔ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉)
106, 9syl6bb 276 . . . . 5 (𝐹:(0...𝐾)⟶𝑉 → (Fun (𝐹 ↾ (1..^𝐾)) ↔ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉))
11 simp-4r 824 . . . . . . . . 9 ((((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ 𝐾 ∈ ℕ0) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → 𝐹:(0...𝐾)⟶𝑉)
12 dff13 6552 . . . . . . . . . . . . . . 15 ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉 ↔ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉 ∧ ∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤)))
13 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑣 = 𝑥 → ((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑥))
1413eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = 𝑥 → (((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤)))
15 equequ1 1998 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = 𝑥 → (𝑣 = 𝑤𝑥 = 𝑤))
1614, 15imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = 𝑥 → ((((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤) ↔ (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑥 = 𝑤)))
17 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 = 𝑦 → ((𝐹 ↾ (1..^𝐾))‘𝑤) = ((𝐹 ↾ (1..^𝐾))‘𝑦))
1817eqeq2d 2661 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 𝑦 → (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
19 equequ2 1999 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 𝑦 → (𝑥 = 𝑤𝑥 = 𝑦))
2018, 19imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑦 → ((((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑥 = 𝑤) ↔ (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦) → 𝑥 = 𝑦)))
2116, 20rspc2va 3354 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) ∧ ∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤)) → (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦) → 𝑥 = 𝑦))
22 fvres 6245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 ∈ (1..^𝐾) → ((𝐹 ↾ (1..^𝐾))‘𝑥) = (𝐹𝑥))
2322eqcomd 2657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 ∈ (1..^𝐾) → (𝐹𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑥))
24 fvres 6245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (1..^𝐾) → ((𝐹 ↾ (1..^𝐾))‘𝑦) = (𝐹𝑦))
2524eqcomd 2657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (1..^𝐾) → (𝐹𝑦) = ((𝐹 ↾ (1..^𝐾))‘𝑦))
2623, 25eqeqan12d 2667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
2726biimpd 219 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
2827imim1d 82 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦) → 𝑥 = 𝑦) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
2928imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) ∧ (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦) → 𝑥 = 𝑦)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
30292a1d 26 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) ∧ (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦) → 𝑥 = 𝑦)) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))
31302a1d 26 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) ∧ (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦) → 𝑥 = 𝑦)) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))))
3231expcom 450 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦) → 𝑥 = 𝑦) → ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))))
3321, 32syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) ∧ ∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤)) → ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))))
3433ex 449 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → (∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤) → ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))))))
3534pm2.43a 54 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → (∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))))
36 ianor 508 . . . . . . . . . . . . . . . . . . . . 21 (¬ (𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) ↔ (¬ 𝑥 ∈ (1..^𝐾) ∨ ¬ 𝑦 ∈ (1..^𝐾)))
37 eqcom 2658 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑦) = (𝐹𝑥))
38 injresinjlem 12628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 𝑥 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑦 ∈ (0...𝐾) ∧ 𝑥 ∈ (0...𝐾)) → ((𝐹𝑦) = (𝐹𝑥) → 𝑦 = 𝑥))))))
3938imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((¬ 𝑥 ∈ (1..^𝐾) ∧ (𝐹‘0) ≠ (𝐹𝐾)) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑦 ∈ (0...𝐾) ∧ 𝑥 ∈ (0...𝐾)) → ((𝐹𝑦) = (𝐹𝑥) → 𝑦 = 𝑥)))))
4039imp41 618 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((¬ 𝑥 ∈ (1..^𝐾) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ (𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0)) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) ∧ (𝑦 ∈ (0...𝐾) ∧ 𝑥 ∈ (0...𝐾))) → ((𝐹𝑦) = (𝐹𝑥) → 𝑦 = 𝑥))
41 eqcom 2658 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = 𝑥𝑥 = 𝑦)
4240, 41syl6ib 241 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((¬ 𝑥 ∈ (1..^𝐾) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ (𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0)) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) ∧ (𝑦 ∈ (0...𝐾) ∧ 𝑥 ∈ (0...𝐾))) → ((𝐹𝑦) = (𝐹𝑥) → 𝑥 = 𝑦))
4337, 42syl5bi 232 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((¬ 𝑥 ∈ (1..^𝐾) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ (𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0)) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) ∧ (𝑦 ∈ (0...𝐾) ∧ 𝑥 ∈ (0...𝐾))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
4443ex 449 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((¬ 𝑥 ∈ (1..^𝐾) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ (𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0)) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → ((𝑦 ∈ (0...𝐾) ∧ 𝑥 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
4544ancomsd 469 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((¬ 𝑥 ∈ (1..^𝐾) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ (𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0)) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
4645exp41 637 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))))
47 injresinjlem 12628 . . . . . . . . . . . . . . . . . . . . . . 23 𝑦 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))))
4846, 47jaoi 393 . . . . . . . . . . . . . . . . . . . . . 22 ((¬ 𝑥 ∈ (1..^𝐾) ∨ ¬ 𝑦 ∈ (1..^𝐾)) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))))
4948a1d 25 . . . . . . . . . . . . . . . . . . . . 21 ((¬ 𝑥 ∈ (1..^𝐾) ∨ ¬ 𝑦 ∈ (1..^𝐾)) → (∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))))
5036, 49sylbi 207 . . . . . . . . . . . . . . . . . . . 20 (¬ (𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → (∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))))
5135, 50pm2.61i 176 . . . . . . . . . . . . . . . . . . 19 (∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))))
5251imp41 618 . . . . . . . . . . . . . . . . . 18 ((((∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ (𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0)) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
5352ralrimivv 2999 . . . . . . . . . . . . . . . . 17 ((((∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ (𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0)) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
5453exp41 637 . . . . . . . . . . . . . . . 16 (∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))
5554adantl 481 . . . . . . . . . . . . . . 15 (((𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉 ∧ ∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤)) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))
5612, 55sylbi 207 . . . . . . . . . . . . . 14 ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉 → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))
5756com13 88 . . . . . . . . . . . . 13 ((𝐹:(0...𝐾)⟶𝑉𝐾 ∈ ℕ0) → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉 → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))
5857ex 449 . . . . . . . . . . . 12 (𝐹:(0...𝐾)⟶𝑉 → (𝐾 ∈ ℕ0 → ((𝐹‘0) ≠ (𝐹𝐾) → ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉 → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))))
5958com24 95 . . . . . . . . . . 11 (𝐹:(0...𝐾)⟶𝑉 → ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉 → ((𝐹‘0) ≠ (𝐹𝐾) → (𝐾 ∈ ℕ0 → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))))
6059impcom 445 . . . . . . . . . 10 (((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉𝐹:(0...𝐾)⟶𝑉) → ((𝐹‘0) ≠ (𝐹𝐾) → (𝐾 ∈ ℕ0 → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))
6160imp41 618 . . . . . . . . 9 ((((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ 𝐾 ∈ ℕ0) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
62 dff13 6552 . . . . . . . . 9 (𝐹:(0...𝐾)–1-1𝑉 ↔ (𝐹:(0...𝐾)⟶𝑉 ∧ ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6311, 61, 62sylanbrc 699 . . . . . . . 8 ((((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ 𝐾 ∈ ℕ0) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → 𝐹:(0...𝐾)–1-1𝑉)
6411biantrurd 528 . . . . . . . . 9 ((((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ 𝐾 ∈ ℕ0) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → (Fun 𝐹 ↔ (𝐹:(0...𝐾)⟶𝑉 ∧ Fun 𝐹)))
65 df-f1 5931 . . . . . . . . 9 (𝐹:(0...𝐾)–1-1𝑉 ↔ (𝐹:(0...𝐾)⟶𝑉 ∧ Fun 𝐹))
6664, 65syl6bbr 278 . . . . . . . 8 ((((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ 𝐾 ∈ ℕ0) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → (Fun 𝐹𝐹:(0...𝐾)–1-1𝑉))
6763, 66mpbird 247 . . . . . . 7 ((((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ 𝐾 ∈ ℕ0) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → Fun 𝐹)
6867ex 449 . . . . . 6 (((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ 𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun 𝐹))
6968exp41 637 . . . . 5 ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉 → (𝐹:(0...𝐾)⟶𝑉 → ((𝐹‘0) ≠ (𝐹𝐾) → (𝐾 ∈ ℕ0 → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun 𝐹)))))
7010, 69syl6bi 243 . . . 4 (𝐹:(0...𝐾)⟶𝑉 → (Fun (𝐹 ↾ (1..^𝐾)) → (𝐹:(0...𝐾)⟶𝑉 → ((𝐹‘0) ≠ (𝐹𝐾) → (𝐾 ∈ ℕ0 → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun 𝐹))))))
7170pm2.43a 54 . . 3 (𝐹:(0...𝐾)⟶𝑉 → (Fun (𝐹 ↾ (1..^𝐾)) → ((𝐹‘0) ≠ (𝐹𝐾) → (𝐾 ∈ ℕ0 → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun 𝐹)))))
72713imp 1275 . 2 ((𝐹:(0...𝐾)⟶𝑉 ∧ Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹‘0) ≠ (𝐹𝐾)) → (𝐾 ∈ ℕ0 → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun 𝐹)))
7372com12 32 1 (𝐾 ∈ ℕ0 → ((𝐹:(0...𝐾)⟶𝑉 ∧ Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹‘0) ≠ (𝐹𝐾)) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  cin 3606  wss 3607  c0 3948  {cpr 4212  ccnv 5142  cres 5145  cima 5146  Fun wfun 5920  wf 5922  1-1wf1 5923  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975  0cn0 11330  ...cfz 12364  ..^cfzo 12504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505
This theorem is referenced by:  pthdepisspth  26687
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