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Theorem injresinj 13894

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) ≠ (𝐹‘𝐾)) → (((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.

Step	Hyp	Ref	Expression
1		fzo0ss1 13792	. . . . . . . . 9 ⊢ (1..^𝐾) ⊆ (0..^𝐾)
2		fzossfz 13781	. . . . . . . . 9 ⊢ (0..^𝐾) ⊆ (0...𝐾)
3	1, 2	sstri 3939	. . . . . . . 8 ⊢ (1..^𝐾) ⊆ (0...𝐾)
4		fssres 6736	. . . . . . . 8 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ (1..^𝐾) ⊆ (0...𝐾)) → (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)
5	3, 4	mpan2 704	. . . . . . 7 ⊢ (𝐹:(0...𝐾)⟶𝑉 → (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)
6	5	biantrud 541	. . . . . 6 ⊢ (𝐹:(0...𝐾)⟶𝑉 → (Fun ^◡(𝐹 ↾ (1..^𝐾)) ↔ (Fun ^◡(𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)))
7		ancom 466	. . . . . . 7 ⊢ ((Fun ^◡(𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉) ↔ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉 ∧ Fun ^◡(𝐹 ↾ (1..^𝐾))))
8		df-f1 6532	. . . . . . 7 ⊢ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 ↔ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉 ∧ Fun ^◡(𝐹 ↾ (1..^𝐾))))
9	7, 8	bitr4i 281	. . . . . 6 ⊢ ((Fun ^◡(𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉) ↔ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉)
10	6, 9	bitrdi 290	. . . . 5 ⊢ (𝐹:(0...𝐾)⟶𝑉 → (Fun ^◡(𝐹 ↾ (1..^𝐾)) ↔ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉))
11		simp-4r 796	. . . . . . . . 9 ⊢ ((((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 ∧ 𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) ∧ 𝐾 ∈ ℕ₀) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → 𝐹:(0...𝐾)⟶𝑉)
12		dff13 7246	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 ↔ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉 ∧ ∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤)))
13		fveqeq2 6882	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = 𝑥 → (((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤)))
14		equequ1 2058	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = 𝑥 → (𝑣 = 𝑤 ↔ 𝑥 = 𝑤))
15	13, 14	imbi12d 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 6873	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ (1..^𝐾))‘𝑤) = ((𝐹 ↾ (1..^𝐾))‘𝑦))
17	16	eqeq2d 2771	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑦 → (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
18		equequ2 2059	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑦 → (𝑥 = 𝑤 ↔ 𝑥 = 𝑦))
19	17, 18	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑦 → ((((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑥 = 𝑤) ↔ (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦) → 𝑥 = 𝑦)))
20	15, 19	rspc2va 3587	. . . . . . . . . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 ∈ (1..^𝐾) → ((𝐹 ↾ (1..^𝐾))‘𝑥) = (𝐹‘𝑥))
22	21	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 ∈ (1..^𝐾) → (𝐹‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑥))
23		fvres 6892	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (1..^𝐾) → ((𝐹 ↾ (1..^𝐾))‘𝑦) = (𝐹‘𝑦))
24	23	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (1..^𝐾) → (𝐹‘𝑦) = ((𝐹 ↾ (1..^𝐾))‘𝑦))
25	22, 24	eqeqan12d 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
26	25	biimpd 232	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
27	26	imim1d 83	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦) → 𝑥 = 𝑦) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
28	27	imp 412	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) ∧ (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦) → 𝑥 = 𝑦)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
29	28	2a1d 27	. . . . . . . . . . . . . . . . . . . . . . . 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)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))
30	29	2a1d 27	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) ∧ (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦) → 𝑥 = 𝑦)) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))))
31	30	expcom 419	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦) → 𝑥 = 𝑦) → ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))))))
32	20, 31	syl 18	. . . . . . . . . . . . . . . . . . . . 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) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))))))
33	32	ex 418	. . . . . . . . . . . . . . . . . . . 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) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))))))
34	33	pm2.43a 55	. . . . . . . . . . . . . . . . . . 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) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))))))
35		ianor 997	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ (𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) ↔ (¬ 𝑥 ∈ (1..^𝐾) ∨ ¬ 𝑦 ∈ (1..^𝐾)))
36		eqcom 2767	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = (𝐹‘𝑥))
37		injresinjlem 13893	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (¬ 𝑥 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑦 ∈ (0...𝐾) ∧ 𝑥 ∈ (0...𝐾)) → ((𝐹‘𝑦) = (𝐹‘𝑥) → 𝑦 = 𝑥))))))
38	37	imp 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((¬ 𝑥 ∈ (1..^𝐾) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑦 ∈ (0...𝐾) ∧ 𝑥 ∈ (0...𝐾)) → ((𝐹‘𝑦) = (𝐹‘𝑥) → 𝑦 = 𝑥)))))
39	38	imp41 431	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((¬ 𝑥 ∈ (1..^𝐾) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) ∧ (𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀)) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) ∧ (𝑦 ∈ (0...𝐾) ∧ 𝑥 ∈ (0...𝐾))) → ((𝐹‘𝑦) = (𝐹‘𝑥) → 𝑦 = 𝑥))
40		eqcom 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
41	39, 40	imbitrdi 254	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((¬ 𝑥 ∈ (1..^𝐾) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) ∧ (𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀)) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) ∧ (𝑦 ∈ (0...𝐾) ∧ 𝑥 ∈ (0...𝐾))) → ((𝐹‘𝑦) = (𝐹‘𝑥) → 𝑥 = 𝑦))
42	36, 41	biimtrid 245	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((¬ 𝑥 ∈ (1..^𝐾) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) ∧ (𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀)) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) ∧ (𝑦 ∈ (0...𝐾) ∧ 𝑥 ∈ (0...𝐾))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
43	42	ex 418	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((¬ 𝑥 ∈ (1..^𝐾) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) ∧ (𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀)) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → ((𝑦 ∈ (0...𝐾) ∧ 𝑥 ∈ (0...𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
44	43	ancomsd 471	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((¬ 𝑥 ∈ (1..^𝐾) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) ∧ (𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀)) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
45	44	exp41 440	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑥 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))))
46		injresinjlem 13893	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑦 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))))
47	45, 46	jaoi 871	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((¬ 𝑥 ∈ (1..^𝐾) ∨ ¬ 𝑦 ∈ (1..^𝐾)) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))))
48	47	a1d 26	. . . . . . . . . . . . . . . . . . . 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) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))))))
49	35, 48	sylbi 220	. . . . . . . . . . . . . . . . . . 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) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))))))
50	34, 49	pm2.61i 184	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))))
51	50	imp41 431	. . . . . . . . . . . . . . . . 17 ⊢ ((((∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) ∧ (𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀)) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → ((𝑥 ∈ (0...𝐾) ∧ 𝑦 ∈ (0...𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
52	51	ralrimivv 3203	. . . . . . . . . . . . . . . 16 ⊢ ((((∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) ∧ (𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀)) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
53	52	exp41 440	. . . . . . . . . . . . . . 15 ⊢ (∀𝑣 ∈ (1..^𝐾)∀𝑤 ∈ (1..^𝐾)(((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) → 𝑣 = 𝑤) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))))
54	12, 53	simplbiim 514	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))))
55	54	com13 89	. . . . . . . . . . . . 13 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ₀) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))))
56	55	ex 418	. . . . . . . . . . . 12 ⊢ (𝐹:(0...𝐾)⟶𝑉 → (𝐾 ∈ ℕ₀ → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))))
57	56	com24 96	. . . . . . . . . . 11 ⊢ (𝐹:(0...𝐾)⟶𝑉 → ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 → ((𝐹‘0) ≠ (𝐹‘𝐾) → (𝐾 ∈ ℕ₀ → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))))
58	57	impcom 413	. . . . . . . . . 10 ⊢ (((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 ∧ 𝐹:(0...𝐾)⟶𝑉) → ((𝐹‘0) ≠ (𝐹‘𝐾) → (𝐾 ∈ ℕ₀ → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))))
59	58	imp41 431	. . . . . . . . 9 ⊢ ((((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 ∧ 𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) ∧ 𝐾 ∈ ℕ₀) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
60		dff13 7246	. . . . . . . . 9 ⊢ (𝐹:(0...𝐾)–1-1→𝑉 ↔ (𝐹:(0...𝐾)⟶𝑉 ∧ ∀𝑥 ∈ (0...𝐾)∀𝑦 ∈ (0...𝐾)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
61	11, 59, 60	sylanbrc 595	. . . . . . . 8 ⊢ ((((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 ∧ 𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) ∧ 𝐾 ∈ ℕ₀) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → 𝐹:(0...𝐾)–1-1→𝑉)
62	11	biantrurd 542	. . . . . . . . 9 ⊢ ((((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 ∧ 𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) ∧ 𝐾 ∈ ℕ₀) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → (Fun ^◡𝐹 ↔ (𝐹:(0...𝐾)⟶𝑉 ∧ Fun ^◡𝐹)))
63		df-f1 6532	. . . . . . . . 9 ⊢ (𝐹:(0...𝐾)–1-1→𝑉 ↔ (𝐹:(0...𝐾)⟶𝑉 ∧ Fun ^◡𝐹))
64	62, 63	bitr4di 292	. . . . . . . 8 ⊢ ((((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 ∧ 𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) ∧ 𝐾 ∈ ℕ₀) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → (Fun ^◡𝐹 ↔ 𝐹:(0...𝐾)–1-1→𝑉))
65	61, 64	mpbird 260	. . . . . . 7 ⊢ ((((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 ∧ 𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) ∧ 𝐾 ∈ ℕ₀) ∧ ((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅) → Fun ^◡𝐹)
66	65	ex 418	. . . . . 6 ⊢ (((((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 ∧ 𝐹:(0...𝐾)⟶𝑉) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) ∧ 𝐾 ∈ ℕ₀) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun ^◡𝐹))
67	66	exp41 440	. . . . 5 ⊢ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 → (𝐹:(0...𝐾)⟶𝑉 → ((𝐹‘0) ≠ (𝐹‘𝐾) → (𝐾 ∈ ℕ₀ → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun ^◡𝐹)))))
68	10, 67	biimtrdi 256	. . . 4 ⊢ (𝐹:(0...𝐾)⟶𝑉 → (Fun ^◡(𝐹 ↾ (1..^𝐾)) → (𝐹:(0...𝐾)⟶𝑉 → ((𝐹‘0) ≠ (𝐹‘𝐾) → (𝐾 ∈ ℕ₀ → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun ^◡𝐹))))))
69	68	pm2.43a 55	. . 3 ⊢ (𝐹:(0...𝐾)⟶𝑉 → (Fun ^◡(𝐹 ↾ (1..^𝐾)) → ((𝐹‘0) ≠ (𝐹‘𝐾) → (𝐾 ∈ ℕ₀ → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun ^◡𝐹)))))
70	69	3imp 1128	. 2 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ Fun ^◡(𝐹 ↾ (1..^𝐾)) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) → (𝐾 ∈ ℕ₀ → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun ^◡𝐹)))
71	70	com12 33	1 ⊢ (𝐾 ∈ ℕ₀ → ((𝐹:(0...𝐾)⟶𝑉 ∧ Fun ^◡(𝐹 ↾ (1..^𝐾)) ∧ (𝐹‘0) ≠ (𝐹‘𝐾)) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → Fun ^◡𝐹)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∩ cin 3897 ⊆ wss 3898 ∅c0 4278 {cpr 4585 ^◡ccnv 5646 ↾ cres 5649 “ cima 5650 Fun wfun 6521 ⟶wf 6523 –1-1→wf1 6524 ‘cfv 6527 (class class class)co 7408 0cc0 11171 1c1 11172 ℕ₀cn0 12575 ...cfz 13608 ..^cfzo 13756

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11227 ax-resscn 11228 ax-1cn 11229 ax-icn 11230 ax-addcl 11231 ax-addrcl 11232 ax-mulcl 11233 ax-mulrcl 11234 ax-mulcom 11235 ax-addass 11236 ax-mulass 11237 ax-distr 11238 ax-i2m1 11239 ax-1ne0 11240 ax-1rid 11241 ax-rnegex 11242 ax-rrecex 11243 ax-cnre 11244 ax-pre-lttri 11245 ax-pre-lttrn 11246 ax-pre-ltadd 11247 ax-pre-mulgt0 11248

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8695 df-en 8952 df-dom 8953 df-sdom 8954 df-pnf 11316 df-mnf 11317 df-xr 11318 df-ltxr 11319 df-le 11320 df-sub 11514 df-neg 11515 df-nn 12305 df-n0 12576 df-z 12663 df-uz 12935 df-fz 13609 df-fzo 13757

This theorem is used by: pthdepisspth 30254

W3C validator