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Theorem injresinj 13150
 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 13059 . . . . . . . . 9 (1..^𝐾) ⊆ (0..^𝐾)
2 fzossfz 13048 . . . . . . . . 9 (0..^𝐾) ⊆ (0...𝐾)
31, 2sstri 3974 . . . . . . . 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 689 . . . . . . 7 (𝐹:(0...𝐾)⟶𝑉 → (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)
65biantrud 534 . . . . . 6 (𝐹:(0...𝐾)⟶𝑉 → (Fun (𝐹 ↾ (1..^𝐾)) ↔ (Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)))
7 ancom 463 . . . . . . 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 280 . . . . . 6 ((Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉) ↔ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉)
106, 9syl6bb 289 . . . . 5 (𝐹:(0...𝐾)⟶𝑉 → (Fun (𝐹 ↾ (1..^𝐾)) ↔ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉))
11 simp-4r 782 . . . . . . . . 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 7005 . . . . . . . . . . . . . . 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 2025 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = 𝑥 → (𝑣 = 𝑤𝑥 = 𝑤))
1513, 14imbi12d 347 . . . . . . . . . . . . . . . . . . . . . . 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 2830 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑦 → (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
18 equequ2 2026 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑦 → (𝑥 = 𝑤𝑥 = 𝑦))
1917, 18imbi12d 347 . . . . . . . . . . . . . . . . . . . . . . 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 3632 . . . . . . . . . . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ (1..^𝐾) → (𝐹𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑥))
23 fvres 6682 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (1..^𝐾) → ((𝐹 ↾ (1..^𝐾))‘𝑦) = (𝐹𝑦))
2423eqcomd 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (1..^𝐾) → (𝐹𝑦) = ((𝐹 ↾ (1..^𝐾))‘𝑦))
2522, 24eqeqan12d 2836 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
2625biimpd 231 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . . . . . . . . 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 977 . . . . . . . . . . . . . . . . . . . 20 (¬ (𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) ↔ (¬ 𝑥 ∈ (1..^𝐾) ∨ ¬ 𝑦 ∈ (1..^𝐾)))
36 eqcom 2826 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑦) = (𝐹𝑥))
37 injresinjlem 13149 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 428 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑥𝑥 = 𝑦)
4139, 40syl6ib 253 . . . . . . . . . . . . . . . . . . . . . . . . . 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 244 . . . . . . . . . . . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . . . . . 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 437 . . . . . . . . . . . . . . . . . . . . . 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 13149 . . . . . . . . . . . . . . . . . . . . . 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 853 . . . . . . . . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . . . . . . . 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 184 . . . . . . . . . . . . . . . . . 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 428 . . . . . . . . . . . . . . . . 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 3188 . . . . . . . . . . . . . . . 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 437 . . . . . . . . . . . . . . 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 507 . . . . . . . . . . . . . 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 415 . . . . . . . . . . . 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 410 . . . . . . . . . 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 428 . . . . . . . . 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 7005 . . . . . . . . 9 (𝐹:(0...𝐾)–1-1𝑉 ↔ (𝐹:(0...𝐾)⟶𝑉 ∧ ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6111, 59, 60sylanbrc 585 . . . . . . . 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 535 . . . . . . . . 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 291 . . . . . . . 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 259 . . . . . . 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 415 . . . . . 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 437 . . . . 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 255 . . . 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 1105 . 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 398   ∨ wo 843   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107   ≠ wne 3014  ∀wral 3136   ∩ cin 3933   ⊆ wss 3934  ∅c0 4289  {cpr 4561  ◡ccnv 5547   ↾ cres 5550   “ cima 5551  Fun wfun 6342  ⟶wf 6344  –1-1→wf1 6345  ‘cfv 6348  (class class class)co 7148  0cc0 10529  1c1 10530  ℕ0cn0 11889  ...cfz 12884  ..^cfzo 13025 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  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 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885  df-fzo 13026 This theorem is referenced by:  pthdepisspth  27508
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