fargshiftf1 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fargshiftf1		Structured version Visualization version GIF version

Theorem fargshiftf1 48135

Description: If a function is 1-1, then also the shifted function is 1-1. (Contributed by Alexander van der Vekens, 23-Nov-2017.)

**Hypothesis**
Ref	Expression
fargshift.g	⊢ 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))

**Assertion**
Ref	Expression
fargshiftf1	⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))–1-1→dom 𝐸)

Distinct variable groups: 𝑥,𝐹 𝑥,𝐸 Allowed substitution hints: 𝐺(𝑥) 𝑁(𝑥)

Proof of Theorem fargshiftf1 Dummy variables 𝑘 𝑙 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1f 6774	. . 3 ⊢ (𝐹:(1...𝑁)–1-1→dom 𝐸 → 𝐹:(1...𝑁)⟶dom 𝐸)
2		fargshift.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
3	2	fargshiftf 48134	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
4	1, 3	sylan2 604	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
5		ffn 6705	. . . . 5 ⊢ (𝐹:(1...𝑁)⟶dom 𝐸 → 𝐹 Fn (1...𝑁))
6		fseq1hash 14411	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)
7	5, 6	sylan2 604	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (♯‘𝐹) = 𝑁)
8	1, 7	sylan2 604	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → (♯‘𝐹) = 𝑁)
9		eleq1 2849	. . . . 5 ⊢ ((♯‘𝐹) = 𝑁 → ((♯‘𝐹) ∈ ℕ₀ ↔ 𝑁 ∈ ℕ₀))
10		oveq2 7418	. . . . . 6 ⊢ ((♯‘𝐹) = 𝑁 → (1...(♯‘𝐹)) = (1...𝑁))
11		f1eq2 6770	. . . . . 6 ⊢ ((1...(♯‘𝐹)) = (1...𝑁) → (𝐹:(1...(♯‘𝐹))–1-1→dom 𝐸 ↔ 𝐹:(1...𝑁)–1-1→dom 𝐸))
12	10, 11	syl 18	. . . . 5 ⊢ ((♯‘𝐹) = 𝑁 → (𝐹:(1...(♯‘𝐹))–1-1→dom 𝐸 ↔ 𝐹:(1...𝑁)–1-1→dom 𝐸))
13	9, 12	anbi12d 643	. . . 4 ⊢ ((♯‘𝐹) = 𝑁 → (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))–1-1→dom 𝐸) ↔ (𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸)))
14		dff13 7252	. . . . . 6 ⊢ (𝐹:(1...(♯‘𝐹))–1-1→dom 𝐸 ↔ (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ∀𝑘 ∈ (1...(♯‘𝐹))∀𝑙 ∈ (1...(♯‘𝐹))((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙)))
15		fz0add1fz1 13763	. . . . . . . . . . 11 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (𝑦 + 1) ∈ (1...(♯‘𝐹)))
16		fz0add1fz1 13763	. . . . . . . . . . 11 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → (𝑧 + 1) ∈ (1...(♯‘𝐹)))
17	15, 16	anim12dan 630	. . . . . . . . . 10 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → ((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹))))
18		fveq2 6881	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑦 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑦 + 1)))
19	18	eqeq1d 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑦 + 1) → ((𝐹‘𝑘) = (𝐹‘𝑙) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘𝑙)))
20		eqeq1 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑦 + 1) → (𝑘 = 𝑙 ↔ (𝑦 + 1) = 𝑙))
21	19, 20	imbi12d 347	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑦 + 1) → (((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙) ↔ ((𝐹‘(𝑦 + 1)) = (𝐹‘𝑙) → (𝑦 + 1) = 𝑙)))
22		fveq2 6881	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = (𝑧 + 1) → (𝐹‘𝑙) = (𝐹‘(𝑧 + 1)))
23	22	eqeq2d 2772	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = (𝑧 + 1) → ((𝐹‘(𝑦 + 1)) = (𝐹‘𝑙) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1))))
24		eqeq2 2773	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = (𝑧 + 1) → ((𝑦 + 1) = 𝑙 ↔ (𝑦 + 1) = (𝑧 + 1)))
25	23, 24	imbi12d 347	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = (𝑧 + 1) → (((𝐹‘(𝑦 + 1)) = (𝐹‘𝑙) → (𝑦 + 1) = 𝑙) ↔ ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1))))
26	21, 25	rspc2v 3591	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹))) → (∀𝑘 ∈ (1...(♯‘𝐹))∀𝑙 ∈ (1...(♯‘𝐹))((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙) → ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1))))
27	26	adantl 486	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹)))) → (∀𝑘 ∈ (1...(♯‘𝐹))∀𝑙 ∈ (1...(♯‘𝐹))((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙) → ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1))))
28	2	fargshiftfv 48133	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → (𝑦 ∈ (0..^(♯‘𝐹)) → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1))))
29	28	expcom 418	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → ((♯‘𝐹) ∈ ℕ₀ → (𝑦 ∈ (0..^(♯‘𝐹)) → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))))
30	29	com13 89	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → ((♯‘𝐹) ∈ ℕ₀ → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))))
31	30	adantr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → ((♯‘𝐹) ∈ ℕ₀ → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))))
32	31	impcom 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1))))
33	32	impcom 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))
34	2	fargshiftfv 48133	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → (𝑧 ∈ (0..^(♯‘𝐹)) → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1))))
35	34	expcom 418	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → ((♯‘𝐹) ∈ ℕ₀ → (𝑧 ∈ (0..^(♯‘𝐹)) → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))))
36	35	com13 89	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ (0..^(♯‘𝐹)) → ((♯‘𝐹) ∈ ℕ₀ → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))))
37	36	adantl 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → ((♯‘𝐹) ∈ ℕ₀ → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))))
38	37	impcom 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1))))
39	38	impcom 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))
40	33, 39	eqeq12d 2777	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → ((𝐺‘𝑦) = (𝐺‘𝑧) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1))))
41	40	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹)))) → ((𝐺‘𝑦) = (𝐺‘𝑧) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1))))
42	41	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹)))) ∧ ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1))) → ((𝐺‘𝑦) = (𝐺‘𝑧) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1))))
43		elfzoelz 13686	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ ℤ)
44	43	zcnd 12700	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ ℂ)
45	44	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → 𝑦 ∈ ℂ)
46	45	adantl 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → 𝑦 ∈ ℂ)
47		elfzoelz 13686	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ (0..^(♯‘𝐹)) → 𝑧 ∈ ℤ)
48	47	zcnd 12700	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∈ (0..^(♯‘𝐹)) → 𝑧 ∈ ℂ)
49	48	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → 𝑧 ∈ ℂ)
50	49	adantl 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → 𝑧 ∈ ℂ)
51		1cnd 11201	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → 1 ∈ ℂ)
52	46, 50, 51	3jca 1144	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ))
53	52	adantl 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ))
54	53	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹)))) → (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ))
55		addcan2 11394	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑦 + 1) = (𝑧 + 1) ↔ 𝑦 = 𝑧))
56	54, 55	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹)))) → ((𝑦 + 1) = (𝑧 + 1) ↔ 𝑦 = 𝑧))
57	56	imbi2d 343	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹)))) → (((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1)) ↔ ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → 𝑦 = 𝑧)))
58	57	biimpa 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹)))) ∧ ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1))) → ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → 𝑦 = 𝑧))
59	42, 58	sylbid 243	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹)))) ∧ ((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1))) → ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧))
60	59	ex 417	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹)))) → (((𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1)) → (𝑦 + 1) = (𝑧 + 1)) → ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧)))
61	27, 60	syld 48	. . . . . . . . . . . . . 14 ⊢ (((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) ∧ ((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹)))) → (∀𝑘 ∈ (1...(♯‘𝐹))∀𝑙 ∈ (1...(♯‘𝐹))((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙) → ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧)))
62	61	exp31 424	. . . . . . . . . . . . 13 ⊢ (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹))) → (∀𝑘 ∈ (1...(♯‘𝐹))∀𝑙 ∈ (1...(♯‘𝐹))((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙) → ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧)))))
63	62	com24 96	. . . . . . . . . . . 12 ⊢ (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (∀𝑘 ∈ (1...(♯‘𝐹))∀𝑙 ∈ (1...(♯‘𝐹))((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙) → (((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹))) → (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧)))))
64	63	imp 411	. . . . . . . . . . 11 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ∀𝑘 ∈ (1...(♯‘𝐹))∀𝑙 ∈ (1...(♯‘𝐹))((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙)) → (((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹))) → (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧))))
65	64	com13 89	. . . . . . . . . 10 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹))) → ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ∀𝑘 ∈ (1...(♯‘𝐹))∀𝑙 ∈ (1...(♯‘𝐹))((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙)) → ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧))))
66	17, 65	mpd 16	. . . . . . . . 9 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ∀𝑘 ∈ (1...(♯‘𝐹))∀𝑙 ∈ (1...(♯‘𝐹))((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙)) → ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧)))
67	66	expcom 418	. . . . . . . 8 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → ((♯‘𝐹) ∈ ℕ₀ → ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ∀𝑘 ∈ (1...(♯‘𝐹))∀𝑙 ∈ (1...(♯‘𝐹))((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙)) → ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧))))
68	67	com13 89	. . . . . . 7 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ∀𝑘 ∈ (1...(♯‘𝐹))∀𝑙 ∈ (1...(♯‘𝐹))((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙)) → ((♯‘𝐹) ∈ ℕ₀ → ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → ((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧))))
69	68	ralrimdvv 3207	. . . . . 6 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ∀𝑘 ∈ (1...(♯‘𝐹))∀𝑙 ∈ (1...(♯‘𝐹))((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙)) → ((♯‘𝐹) ∈ ℕ₀ → ∀𝑦 ∈ (0..^(♯‘𝐹))∀𝑧 ∈ (0..^(♯‘𝐹))((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧)))
70	14, 69	sylbi 220	. . . . 5 ⊢ (𝐹:(1...(♯‘𝐹))–1-1→dom 𝐸 → ((♯‘𝐹) ∈ ℕ₀ → ∀𝑦 ∈ (0..^(♯‘𝐹))∀𝑧 ∈ (0..^(♯‘𝐹))((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧)))
71	70	impcom 412	. . . 4 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))–1-1→dom 𝐸) → ∀𝑦 ∈ (0..^(♯‘𝐹))∀𝑧 ∈ (0..^(♯‘𝐹))((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧))
72	13, 71	biimtrrdi 257	. . 3 ⊢ ((♯‘𝐹) = 𝑁 → ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → ∀𝑦 ∈ (0..^(♯‘𝐹))∀𝑧 ∈ (0..^(♯‘𝐹))((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧)))
73	8, 72	mpcom 39	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → ∀𝑦 ∈ (0..^(♯‘𝐹))∀𝑧 ∈ (0..^(♯‘𝐹))((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧))
74		dff13 7252	. 2 ⊢ (𝐺:(0..^(♯‘𝐹))–1-1→dom 𝐸 ↔ (𝐺:(0..^(♯‘𝐹))⟶dom 𝐸 ∧ ∀𝑦 ∈ (0..^(♯‘𝐹))∀𝑧 ∈ (0..^(♯‘𝐹))((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧)))
75	4, 73, 74	sylanbrc 594	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))–1-1→dom 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 ∀wral 3077 ↦ cmpt 5191 dom cdm 5661 Fn wfn 6531 ⟶wf 6532 –1-1→wf1 6533 ‘cfv 6536 (class class class)co 7410 ℂcc 11097 0cc0 11099 1c1 11100 + caddc 11102 ℕ₀cn0 12503 ...cfz 13534 ..^cfzo 13681 ♯chash 14365

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-n0 12504 df-z 12591 df-uz 12862 df-fz 13535 df-fzo 13682 df-hash 14366

This theorem is referenced by: (None)

W3C validator