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Theorem injresinj 13753
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 13662 . . . . . . . . 9 (1..^𝐾) ⊆ (0..^𝐾)
2 fzossfz 13651 . . . . . . . . 9 (0..^𝐾) ⊆ (0...𝐾)
31, 2sstri 3992 . . . . . . . 8 (1..^𝐾) ⊆ (0...𝐾)
4 fssres 6758 . . . . . . . 8 ((𝐹:(0...𝐾)⟶𝑉 ∧ (1..^𝐾) ⊆ (0...𝐾)) → (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)
53, 4mpan2 690 . . . . . . 7 (𝐹:(0...𝐾)⟶𝑉 → (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)
65biantrud 533 . . . . . 6 (𝐹:(0...𝐾)⟶𝑉 → (Fun (𝐹 ↾ (1..^𝐾)) ↔ (Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)))
7 ancom 462 . . . . . . 7 ((Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉) ↔ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉 ∧ Fun (𝐹 ↾ (1..^𝐾))))
8 df-f1 6549 . . . . . . 7 ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉 ↔ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉 ∧ Fun (𝐹 ↾ (1..^𝐾))))
97, 8bitr4i 278 . . . . . 6 ((Fun (𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉) ↔ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉)
106, 9bitrdi 287 . . . . 5 (𝐹:(0...𝐾)⟶𝑉 → (Fun (𝐹 ↾ (1..^𝐾)) ↔ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1𝑉))
11 simp-4r 783 . . . . . . . . 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 7254 . . . . . . . . . . . . . . 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 6901 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = 𝑥 → (((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤)))
14 equequ1 2029 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = 𝑥 → (𝑣 = 𝑤𝑥 = 𝑤))
1513, 14imbi12d 345 . . . . . . . . . . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 𝑦 → ((𝐹 ↾ (1..^𝐾))‘𝑤) = ((𝐹 ↾ (1..^𝐾))‘𝑦))
1716eqeq2d 2744 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑦 → (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
18 equequ2 2030 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑦 → (𝑥 = 𝑤𝑥 = 𝑦))
1917, 18imbi12d 345 . . . . . . . . . . . . . . . . . . . . . . 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 3624 . . . . . . . . . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 ∈ (1..^𝐾) → ((𝐹 ↾ (1..^𝐾))‘𝑥) = (𝐹𝑥))
2221eqcomd 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 ∈ (1..^𝐾) → (𝐹𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑥))
23 fvres 6911 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (1..^𝐾) → ((𝐹 ↾ (1..^𝐾))‘𝑦) = (𝐹𝑦))
2423eqcomd 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (1..^𝐾) → (𝐹𝑦) = ((𝐹 ↾ (1..^𝐾))‘𝑦))
2522, 24eqeqan12d 2747 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
2625biimpd 228 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . . . . . . 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 981 . . . . . . . . . . . . . . . . . . . 20 (¬ (𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) ↔ (¬ 𝑥 ∈ (1..^𝐾) ∨ ¬ 𝑦 ∈ (1..^𝐾)))
36 eqcom 2740 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹𝑦) = (𝐹𝑥))
37 injresinjlem 13752 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 427 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2740 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑥𝑥 = 𝑦)
4139, 40imbitrdi 250 . . . . . . . . . . . . . . . . . . . . . . . . . 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, 41biimtrid 241 . . . . . . . . . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . . . . . . . . . . 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 13752 . . . . . . . . . . . . . . . . . . . . . 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 856 . . . . . . . . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . . . . . . . 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 182 . . . . . . . . . . . . . . . . . 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 427 . . . . . . . . . . . . . . . . 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 3199 . . . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . . . 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 506 . . . . . . . . . . . . . 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 414 . . . . . . . . . . . 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 409 . . . . . . . . . 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 427 . . . . . . . . 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 7254 . . . . . . . . 9 (𝐹:(0...𝐾)–1-1𝑉 ↔ (𝐹:(0...𝐾)⟶𝑉 ∧ ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6111, 59, 60sylanbrc 584 . . . . . . . 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 534 . . . . . . . . 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 6549 . . . . . . . . 9 (𝐹:(0...𝐾)–1-1𝑉 ↔ (𝐹:(0...𝐾)⟶𝑉 ∧ Fun 𝐹))
6462, 63bitr4di 289 . . . . . . . 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 257 . . . . . . 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 414 . . . . . 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 436 . . . . 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 253 . . . 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 1112 . 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 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  cin 3948  wss 3949  c0 4323  {cpr 4631  ccnv 5676  cres 5679  cima 5680  Fun wfun 6538  wf 6540  1-1wf1 6541  cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111  0cn0 12472  ...cfz 13484  ..^cfzo 13627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628
This theorem is referenced by:  pthdepisspth  28992
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