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Theorem injresinj 13789

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 13697	. . . . . . . . 9 ⊢ (1..^𝐾) ⊆ (0..^𝐾)
2		fzossfz 13686	. . . . . . . . 9 ⊢ (0..^𝐾) ⊆ (0...𝐾)
3	1, 2	sstri 3986	. . . . . . . 8 ⊢ (1..^𝐾) ⊆ (0...𝐾)
4		fssres 6763	. . . . . . . 8 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ (1..^𝐾) ⊆ (0...𝐾)) → (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)
5	3, 4	mpan2 689	. . . . . . 7 ⊢ (𝐹:(0...𝐾)⟶𝑉 → (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)
6	5	biantrud 530	. . . . . 6 ⊢ (𝐹:(0...𝐾)⟶𝑉 → (Fun ^◡(𝐹 ↾ (1..^𝐾)) ↔ (Fun ^◡(𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)))
7		ancom 459	. . . . . . 7 ⊢ ((Fun ^◡(𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉) ↔ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉 ∧ Fun ^◡(𝐹 ↾ (1..^𝐾))))
8		df-f1 6554	. . . . . . 7 ⊢ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉 ↔ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉 ∧ Fun ^◡(𝐹 ↾ (1..^𝐾))))
9	7, 8	bitr4i 277	. . . . . 6 ⊢ ((Fun ^◡(𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉) ↔ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉)
10	6, 9	bitrdi 286	. . . . 5 ⊢ (𝐹:(0...𝐾)⟶𝑉 → (Fun ^◡(𝐹 ↾ (1..^𝐾)) ↔ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉))
11		simp-4r 782	. . . . . . . . 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 7265	. . . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = 𝑥 → (((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤)))
14		equequ1 2020	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = 𝑥 → (𝑣 = 𝑤 ↔ 𝑥 = 𝑤))
15	13, 14	imbi12d 343	. . . . . . . . . . . . . . . . . . . . . . 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 6896	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ (1..^𝐾))‘𝑤) = ((𝐹 ↾ (1..^𝐾))‘𝑦))
17	16	eqeq2d 2736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑦 → (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
18		equequ2 2021	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑦 → (𝑥 = 𝑤 ↔ 𝑥 = 𝑦))
19	17, 18	imbi12d 343	. . . . . . . . . . . . . . . . . . . . . . 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 3618	. . . . . . . . . . . . . . . . . . . . . 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 6915	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 ∈ (1..^𝐾) → ((𝐹 ↾ (1..^𝐾))‘𝑥) = (𝐹‘𝑥))
22	21	eqcomd 2731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 ∈ (1..^𝐾) → (𝐹‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑥))
23		fvres 6915	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (1..^𝐾) → ((𝐹 ↾ (1..^𝐾))‘𝑦) = (𝐹‘𝑦))
24	23	eqcomd 2731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (1..^𝐾) → (𝐹‘𝑦) = ((𝐹 ↾ (1..^𝐾))‘𝑦))
25	22, 24	eqeqan12d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑥 ∈ (1..^𝐾) ∧ 𝑦 ∈ (1..^𝐾)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
26	25	biimpd 228	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = (𝐹‘𝑥))
37		injresinjlem 13788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 424	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
41	39, 40	imbitrdi 250	. . . . . . . . . . . . . . . . . . . . . . . . . 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 241	. . . . . . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . . . . . 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 464	. . . . . . . . . . . . . . . . . . . . . . 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 433	. . . . . . . . . . . . . . . . . . . . . 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 13788	. . . . . . . . . . . . . . . . . . . . . 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 855	. . . . . . . . . . . . . . . . . . . . 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 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) ≠ (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 182	. . . . . . . . . . . . . . . . . 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 424	. . . . . . . . . . . . . . . . 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 3188	. . . . . . . . . . . . . . . 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 433	. . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . . 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 411	. . . . . . . . . . . 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 406	. . . . . . . . . 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 424	. . . . . . . . 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 7265	. . . . . . . . 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 581	. . . . . . . 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 531	. . . . . . . . 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 6554	. . . . . . . . 9 ⊢ (𝐹:(0...𝐾)–1-1→𝑉 ↔ (𝐹:(0...𝐾)⟶𝑉 ∧ Fun ^◡𝐹))
64	62, 63	bitr4di 288	. . . . . . . 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 256	. . . . . . 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 411	. . . . . 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 433	. . . . 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 252	. . . 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 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 ∩ cin 3943 ⊆ wss 3944 ∅c0 4322 {cpr 4632 ^◡ccnv 5677 ↾ cres 5680 “ cima 5681 Fun wfun 6543 ⟶wf 6545 –1-1→wf1 6546 ‘cfv 6549 (class class class)co 7419 0cc0 11140 1c1 11141 ℕ₀cn0 12505 ...cfz 13519 ..^cfzo 13662

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 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663

This theorem is referenced by: pthdepisspth 29621

W3C validator