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Theorem injresinj 13153

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 13062	. . . . . . . . 9 ⊢ (1..^𝐾) ⊆ (0..^𝐾)
2		fzossfz 13051	. . . . . . . . 9 ⊢ (0..^𝐾) ⊆ (0...𝐾)
3	1, 2	sstri 3924	. . . . . . . 8 ⊢ (1..^𝐾) ⊆ (0...𝐾)
4		fssres 6518	. . . . . . . 8 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ (1..^𝐾) ⊆ (0...𝐾)) → (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)
5	3, 4	mpan2 690	. . . . . . 7 ⊢ (𝐹:(0...𝐾)⟶𝑉 → (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)
6	5	biantrud 535	. . . . . 6 ⊢ (𝐹:(0...𝐾)⟶𝑉 → (Fun ^◡(𝐹 ↾ (1..^𝐾)) ↔ (Fun ^◡(𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)))
7		ancom 464	. . . . . . 7 ⊢ ((Fun ^◡(𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉) ↔ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉 ∧ Fun ^◡(𝐹 ↾ (1..^𝐾))))
8		df-f1 6329	. . . . . . 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	syl6bb 290	. . . . 5 ⊢ (𝐹:(0...𝐾)⟶𝑉 → (Fun ^◡(𝐹 ↾ (1..^𝐾)) ↔ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉))
11		simp-4r 783	. . . . . . . . 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 6991	. . . . . . . . . . . . . . 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 6654	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = 𝑥 → (((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤)))
14		equequ1 2032	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = 𝑥 → (𝑣 = 𝑤 ↔ 𝑥 = 𝑤))
15	13, 14	imbi12d 348	. . . . . . . . . . . . . . . . . . . . . . 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 6645	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ (1..^𝐾))‘𝑤) = ((𝐹 ↾ (1..^𝐾))‘𝑦))
17	16	eqeq2d 2809	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑦 → (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
18		equequ2 2033	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑦 → (𝑥 = 𝑤 ↔ 𝑥 = 𝑦))
19	17, 18	imbi12d 348	. . . . . . . . . . . . . . . . . . . . . . 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 3582	. . . . . . . . . . . . . . . . . . . . . 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 6664	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 ∈ (1..^𝐾) → ((𝐹 ↾ (1..^𝐾))‘𝑥) = (𝐹‘𝑥))
22	21	eqcomd 2804	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 ∈ (1..^𝐾) → (𝐹‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑥))
23		fvres 6664	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (1..^𝐾) → ((𝐹 ↾ (1..^𝐾))‘𝑦) = (𝐹‘𝑦))
24	23	eqcomd 2804	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (1..^𝐾) → (𝐹‘𝑦) = ((𝐹 ↾ (1..^𝐾))‘𝑦))
25	22, 24	eqeqan12d 2815	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 82	. . . . . . . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . . . . . . 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 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)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))
30	29	2a1d 26	. . . . . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . . . . 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 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) ≠ (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 416	. . . . . . . . . . . . . . . . . . . 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 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) ≠ (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 979	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ (𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) ↔ (¬ 𝑥 ∈ (1..^𝐾) ∨ ¬ 𝑦 ∈ (1..^𝐾)))
36		eqcom 2805	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = (𝐹‘𝑥))
37		injresinjlem 13152	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 429	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
41	39, 40	syl6ib 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	syl5bi 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 416	. . . . . . . . . . . . . . . . . . . . . . . 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 469	. . . . . . . . . . . . . . . . . . . . . . 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 438	. . . . . . . . . . . . . . . . . . . . . 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 13152	. . . . . . . . . . . . . . . . . . . . . 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 854	. . . . . . . . . . . . . . . . . . . . 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 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) ≠ (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 185	. . . . . . . . . . . . . . . . . 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 429	. . . . . . . . . . . . . . . . 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 3155	. . . . . . . . . . . . . . . 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 438	. . . . . . . . . . . . . . 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 508	. . . . . . . . . . . . . 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 88	. . . . . . . . . . . . 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 416	. . . . . . . . . . . 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 95	. . . . . . . . . . 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 411	. . . . . . . . . 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 429	. . . . . . . . 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 6991	. . . . . . . . 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 586	. . . . . . . 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 536	. . . . . . . . 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 6329	. . . . . . . . 9 ⊢ (𝐹:(0...𝐾)–1-1→𝑉 ↔ (𝐹:(0...𝐾)⟶𝑉 ∧ Fun ^◡𝐹))
64	62, 63	syl6bbr 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 416	. . . . . 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 438	. . . . 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	syl6bi 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 54	. . 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 1108	. 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 32	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

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {cpr 4527 ^◡ccnv 5518 ↾ cres 5521 “ cima 5522 Fun wfun 6318 ⟶wf 6320 –1-1→wf1 6321 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 ℕ₀cn0 11885 ...cfz 12885 ..^cfzo 13028

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029

This theorem is referenced by: pthdepisspth 27524

W3C validator