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Theorem injresinj 13146
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 13055 . . . . . . . . 9 (1..^𝐾) ⊆ (0..^𝐾)
2 fzossfz 13044 . . . . . . . . 9 (0..^𝐾) ⊆ (0...𝐾)
31, 2sstri 3973 . . . . . . . 8 (1..^𝐾) ⊆ (0...𝐾)
4 fssres 6537 . . . . . . . 8 ((𝐹:(0...𝐾)⟶𝑉 ∧ (1..^𝐾) ⊆ (0...𝐾)) → (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)
53, 4mpan2 687 . . . . . . 7 (𝐹:(0...𝐾)⟶𝑉 → (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)
65biantrud 532 . . . . . 6 (𝐹:(0...𝐾)⟶𝑉 → (Fun (𝐹 ↾ (1..^𝐾)) ↔ (Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)))
7 ancom 461 . . . . . . 7 ((Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉) ↔ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉 ∧ Fun (𝐹 ↾ (1..^𝐾))))
8 df-f1 6353 . . . . . . 7 ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉 ↔ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉 ∧ Fun (𝐹 ↾ (1..^𝐾))))
97, 8bitr4i 279 . . . . . 6 ((Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉) ↔ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉)
106, 9syl6bb 288 . . . . 5 (𝐹:(0...𝐾)⟶𝑉 → (Fun (𝐹 ↾ (1..^𝐾)) ↔ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉))
11 simp-4r 780 . . . . . . . . 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 7004 . . . . . . . . . . . . . . 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 fveqeq2 6672 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = 𝑥 → (((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤)))
14 equequ1 2023 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = 𝑥 → (𝑣 = 𝑤𝑥 = 𝑤))
1513, 14imbi12d 346 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = 𝑥 → ((((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤) ↔ (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑥 = 𝑤)))
16 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 𝑦 → ((𝐹 ↾ (1..^𝐾))‘𝑤) = ((𝐹 ↾ (1..^𝐾))‘𝑦))
1716eqeq2d 2829 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑦 → (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
18 equequ2 2024 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑦 → (𝑥 = 𝑤𝑥 = 𝑦))
1917, 18imbi12d 346 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑦 → ((((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑥 = 𝑤) ↔ (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦) → 𝑥 = 𝑦)))
2015, 19rspc2va 3631 . . . . . . . . . . . . . . . . . . . . . 22 (((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))
21 fvres 6682 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 ∈ (1..^𝐾) → ((𝐹 ↾ (1..^𝐾))‘𝑥) = (𝐹𝑥))
2221eqcomd 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ (1..^𝐾) → (𝐹𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑥))
23 fvres 6682 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (1..^𝐾) → ((𝐹 ↾ (1..^𝐾))‘𝑦) = (𝐹𝑦))
2423eqcomd 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (1..^𝐾) → (𝐹𝑦) = ((𝐹 ↾ (1..^𝐾))‘𝑦))
2522, 24eqeqan12d 2835 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
2625biimpd 230 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((𝐹𝑥) = (𝐹𝑦) → ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
2726imim1d 82 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦) → 𝑥 = 𝑦) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
2827imp 407 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) ∧ (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦) → 𝑥 = 𝑦)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
29282a1d 26 . . . . . . . . . . . . . . . . . . . . . . . 24 (((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))))
30292a1d 26 . . . . . . . . . . . . . . . . . . . . . . 23 (((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))))))
3130expcom 414 . . . . . . . . . . . . . . . . . . . . . 22 ((((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)))))))
3220, 31syl 17 . . . . . . . . . . . . . . . . . . . . 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)))))))
3332ex 413 . . . . . . . . . . . . . . . . . . . 20 ((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))))))))
3433pm2.43a 54 . . . . . . . . . . . . . . . . . . 19 ((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)))))))
35 ianor 975 . . . . . . . . . . . . . . . . . . . 20 (¬ (𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) ↔ (¬ 𝑥 ∈ (1..^𝐾) ∨ ¬ 𝑦 ∈ (1..^𝐾)))
36 eqcom 2825 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑦) = (𝐹𝑥))
37 injresinjlem 13145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
3837imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
3938imp41 426 . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → 𝑦 = 𝑥))
40 eqcom 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑥𝑥 = 𝑦)
4139, 40syl6ib 252 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((¬ 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))
4236, 41syl5bi 243 . . . . . . . . . . . . . . . . . . . . . . . . 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))
4342ex 413 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((¬ 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)))
4443ancomsd 466 . . . . . . . . . . . . . . . . . . . . . . 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)))
4544exp41 435 . . . . . . . . . . . . . . . . . . . . . 22 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))))))
46 injresinjlem 13145 . . . . . . . . . . . . . . . . . . . . . 22 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))))))
4745, 46jaoi 851 . . . . . . . . . . . . . . . . . . . . 21 ((¬ 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))))))
4847a1d 25 . . . . . . . . . . . . . . . . . . . 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)))))))
4935, 48sylbi 218 . . . . . . . . . . . . . . . . . . 19 (¬ (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)))))))
5034, 49pm2.61i 183 . . . . . . . . . . . . . . . . . 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))))))
5150imp41 426 . . . . . . . . . . . . . . . . 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)))
5251ralrimivv 3187 . . . . . . . . . . . . . . . 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))
5352exp41 435 . . . . . . . . . . . . . . 15 (∀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)))))
5412, 53simplbiim 505 . . . . . . . . . . . . . 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)))))
5554com13 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)))))
5655ex 413 . . . . . . . . . . . 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))))))
5756com24 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))))))
5857impcom 408 . . . . . . . . . 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)))))
5958imp41 426 . . . . . . . . 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))
60 dff13 7004 . . . . . . . . 9 (𝐹:(0...𝐾)–1-1𝑉 ↔ (𝐹:(0...𝐾)⟶𝑉 ∧ ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6111, 59, 60sylanbrc 583 . . . . . . . 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)
6211biantrurd 533 . . . . . . . . 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)))
63 df-f1 6353 . . . . . . . . 9 (𝐹:(0...𝐾)–1-1𝑉 ↔ (𝐹:(0...𝐾)⟶𝑉 ∧ Fun 𝐹))
6462, 63syl6bbr 290 . . . . . . . 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))
6561, 64mpbird 258 . . . . . . 7 ((((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ 𝐾 ∈ ℕ0) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → Fun 𝐹)
6665ex 413 . . . . . 6 (((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹𝐾)) ∧ 𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun 𝐹))
6766exp41 435 . . . . 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)))))
6810, 67syl6bi 254 . . . 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))))))
6968pm2.43a 54 . . 3 (𝐹:(0...𝐾)⟶𝑉 → (Fun (𝐹 ↾ (1..^𝐾)) → ((𝐹‘0) ≠ (𝐹𝐾) → (𝐾 ∈ ℕ0 → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun 𝐹)))))
70693imp 1103 . 2 ((𝐹:(0...𝐾)⟶𝑉 ∧ Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹‘0) ≠ (𝐹𝐾)) → (𝐾 ∈ ℕ0 → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun 𝐹)))
7170com12 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  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  cin 3932  wss 3933  c0 4288  {cpr 4559  ccnv 5547  cres 5550  cima 5551  Fun wfun 6342  wf 6344  1-1wf1 6345  cfv 6348  (class class class)co 7145  0cc0 10525  1c1 10526  0cn0 11885  ...cfz 12880  ..^cfzo 13021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022
This theorem is referenced by:  pthdepisspth  27443
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