injresinj - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > injresinj		Structured version Visualization version GIF version

Theorem injresinj 13508

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 13417	. . . . . . . . 9 ⊢ (1..^𝐾) ⊆ (0..^𝐾)
2		fzossfz 13406	. . . . . . . . 9 ⊢ (0..^𝐾) ⊆ (0...𝐾)
3	1, 2	sstri 3930	. . . . . . . 8 ⊢ (1..^𝐾) ⊆ (0...𝐾)
4		fssres 6640	. . . . . . . 8 ⊢ ((𝐹:(0...𝐾)⟶𝑉 ∧ (1..^𝐾) ⊆ (0...𝐾)) → (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)
5	3, 4	mpan2 688	. . . . . . 7 ⊢ (𝐹:(0...𝐾)⟶𝑉 → (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)
6	5	biantrud 532	. . . . . 6 ⊢ (𝐹:(0...𝐾)⟶𝑉 → (Fun ^◡(𝐹 ↾ (1..^𝐾)) ↔ (Fun ^◡(𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉)))
7		ancom 461	. . . . . . 7 ⊢ ((Fun ^◡(𝐹 ↾ (1..^𝐾)) ∧ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉) ↔ ((𝐹 ↾ (1..^𝐾)):(1..^𝐾)⟶𝑉 ∧ Fun ^◡(𝐹 ↾ (1..^𝐾))))
8		df-f1 6438	. . . . . . 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 287	. . . . 5 ⊢ (𝐹:(0...𝐾)⟶𝑉 → (Fun ^◡(𝐹 ↾ (1..^𝐾)) ↔ (𝐹 ↾ (1..^𝐾)):(1..^𝐾)–1-1→𝑉))
11		simp-4r 781	. . . . . . . . 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 7128	. . . . . . . . . . . . . . 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 6783	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = 𝑥 → (((𝐹 ↾ (1..^𝐾))‘𝑣) = ((𝐹 ↾ (1..^𝐾))‘𝑤) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤)))
14		equequ1 2028	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = 𝑥 → (𝑣 = 𝑤 ↔ 𝑥 = 𝑤))
15	13, 14	imbi12d 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 6774	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = 𝑦 → ((𝐹 ↾ (1..^𝐾))‘𝑤) = ((𝐹 ↾ (1..^𝐾))‘𝑦))
17	16	eqeq2d 2749	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑦 → (((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑤) ↔ ((𝐹 ↾ (1..^𝐾))‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑦)))
18		equequ2 2029	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑦 → (𝑥 = 𝑤 ↔ 𝑥 = 𝑦))
19	17, 18	imbi12d 345	. . . . . . . . . . . . . . . . . . . . . . 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 3571	. . . . . . . . . . . . . . . . . . . . . 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 6793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 ∈ (1..^𝐾) → ((𝐹 ↾ (1..^𝐾))‘𝑥) = (𝐹‘𝑥))
22	21	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 ∈ (1..^𝐾) → (𝐹‘𝑥) = ((𝐹 ↾ (1..^𝐾))‘𝑥))
23		fvres 6793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (1..^𝐾) → ((𝐹 ↾ (1..^𝐾))‘𝑦) = (𝐹‘𝑦))
24	23	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (1..^𝐾) → (𝐹‘𝑦) = ((𝐹 ↾ (1..^𝐾))‘𝑦))
25	22, 24	eqeqan12d 2752	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = (𝐹‘𝑥))
37		injresinjlem 13507	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 426	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
41	39, 40	syl6ib 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	syl5bi 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 413	. . . . . . . . . . . . . . . . . . . . . . . 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 466	. . . . . . . . . . . . . . . . . . . . . . 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 435	. . . . . . . . . . . . . . . . . . . . . 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 13507	. . . . . . . . . . . . . . . . . . . . . 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 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 426	. . . . . . . . . . . . . . . . 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 3122	. . . . . . . . . . . . . . . 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 435	. . . . . . . . . . . . . . 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 505	. . . . . . . . . . . . . 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 413	. . . . . . . . . . . 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 408	. . . . . . . . . 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 426	. . . . . . . . 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 7128	. . . . . . . . 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 583	. . . . . . . 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 533	. . . . . . . . 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 6438	. . . . . . . . 9 ⊢ (𝐹:(0...𝐾)–1-1→𝑉 ↔ (𝐹:(0...𝐾)⟶𝑉 ∧ Fun ^◡𝐹))
64	62, 63	bitr4di 289	. . . . . . . 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 413	. . . . . 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 435	. . . . 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 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 1110	. 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 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 {cpr 4563 ^◡ccnv 5588 ↾ cres 5591 “ cima 5592 Fun wfun 6427 ⟶wf 6429 –1-1→wf1 6430 ‘cfv 6433 (class class class)co 7275 0cc0 10871 1c1 10872 ℕ₀cn0 12233 ...cfz 13239 ..^cfzo 13382

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383

This theorem is referenced by: pthdepisspth 28103

W3C validator