fargshiftf1 - Mathbox for Alexander van der Vekens

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Theorem fargshiftf1 44520

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 6604	. . 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 44519	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
4	1, 3	sylan2 596	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
5		ffn 6534	. . . . 5 ⊢ (𝐹:(1...𝑁)⟶dom 𝐸 → 𝐹 Fn (1...𝑁))
6		fseq1hash 13926	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)
7	5, 6	sylan2 596	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (♯‘𝐹) = 𝑁)
8	1, 7	sylan2 596	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → (♯‘𝐹) = 𝑁)
9		eleq1 2821	. . . . 5 ⊢ ((♯‘𝐹) = 𝑁 → ((♯‘𝐹) ∈ ℕ₀ ↔ 𝑁 ∈ ℕ₀))
10		oveq2 7210	. . . . . 6 ⊢ ((♯‘𝐹) = 𝑁 → (1...(♯‘𝐹)) = (1...𝑁))
11		f1eq2 6600	. . . . . 6 ⊢ ((1...(♯‘𝐹)) = (1...𝑁) → (𝐹:(1...(♯‘𝐹))–1-1→dom 𝐸 ↔ 𝐹:(1...𝑁)–1-1→dom 𝐸))
12	10, 11	syl 17	. . . . 5 ⊢ ((♯‘𝐹) = 𝑁 → (𝐹:(1...(♯‘𝐹))–1-1→dom 𝐸 ↔ 𝐹:(1...𝑁)–1-1→dom 𝐸))
13	9, 12	anbi12d 634	. . . 4 ⊢ ((♯‘𝐹) = 𝑁 → (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))–1-1→dom 𝐸) ↔ (𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸)))
14		dff13 7056	. . . . . 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 13295	. . . . . . . . . . 11 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (𝑦 + 1) ∈ (1...(♯‘𝐹)))
16		fz0add1fz1 13295	. . . . . . . . . . 11 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → (𝑧 + 1) ∈ (1...(♯‘𝐹)))
17	15, 16	anim12dan 622	. . . . . . . . . 10 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → ((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹))))
18		fveq2 6706	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑦 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑦 + 1)))
19	18	eqeq1d 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑦 + 1) → ((𝐹‘𝑘) = (𝐹‘𝑙) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘𝑙)))
20		eqeq1 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑦 + 1) → (𝑘 = 𝑙 ↔ (𝑦 + 1) = 𝑙))
21	19, 20	imbi12d 348	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑦 + 1) → (((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙) ↔ ((𝐹‘(𝑦 + 1)) = (𝐹‘𝑙) → (𝑦 + 1) = 𝑙)))
22		fveq2 6706	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = (𝑧 + 1) → (𝐹‘𝑙) = (𝐹‘(𝑧 + 1)))
23	22	eqeq2d 2745	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = (𝑧 + 1) → ((𝐹‘(𝑦 + 1)) = (𝐹‘𝑙) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1))))
24		eqeq2 2746	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = (𝑧 + 1) → ((𝑦 + 1) = 𝑙 ↔ (𝑦 + 1) = (𝑧 + 1)))
25	23, 24	imbi12d 348	. . . . . . . . . . . . . . . . 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 3540	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 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 44518	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → (𝑦 ∈ (0..^(♯‘𝐹)) → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1))))
29	28	expcom 417	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → ((♯‘𝐹) ∈ ℕ₀ → (𝑦 ∈ (0..^(♯‘𝐹)) → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))))
30	29	com13 88	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → ((♯‘𝐹) ∈ ℕ₀ → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))))
31	30	adantr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → ((♯‘𝐹) ∈ ℕ₀ → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))))
32	31	impcom 411	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1))))
33	32	impcom 411	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))
34	2	fargshiftfv 44518	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → (𝑧 ∈ (0..^(♯‘𝐹)) → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1))))
35	34	expcom 417	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → ((♯‘𝐹) ∈ ℕ₀ → (𝑧 ∈ (0..^(♯‘𝐹)) → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))))
36	35	com13 88	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ (0..^(♯‘𝐹)) → ((♯‘𝐹) ∈ ℕ₀ → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))))
37	36	adantl 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → ((♯‘𝐹) ∈ ℕ₀ → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))))
38	37	impcom 411	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1))))
39	38	impcom 411	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))
40	33, 39	eqeq12d 2750	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . 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 13226	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ ℤ)
44	43	zcnd 12266	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ ℂ)
45	44	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → 𝑦 ∈ ℂ)
46	45	adantl 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → 𝑦 ∈ ℂ)
47		elfzoelz 13226	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ (0..^(♯‘𝐹)) → 𝑧 ∈ ℤ)
48	47	zcnd 12266	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∈ (0..^(♯‘𝐹)) → 𝑧 ∈ ℂ)
49	48	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → 𝑧 ∈ ℂ)
50	49	adantl 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → 𝑧 ∈ ℂ)
51		1cnd 10811	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → 1 ∈ ℂ)
52	46, 50, 51	3jca 1130	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ))
53	52	adantl 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ))
54	53	adantr 484	. . . . . . . . . . . . . . . . . . . 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 11000	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑦 + 1) = (𝑧 + 1) ↔ 𝑦 = 𝑧))
56	54, 55	syl 17	. . . . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . 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 47	. . . . . . . . . . . . . 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 423	. . . . . . . . . . . . 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 95	. . . . . . . . . . . 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 410	. . . . . . . . . . 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 88	. . . . . . . . . 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 15	. . . . . . . . 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 417	. . . . . . . 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 88	. . . . . . 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 3107	. . . . . 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 411	. . . 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	syl6bir 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 38	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → ∀𝑦 ∈ (0..^(♯‘𝐹))∀𝑧 ∈ (0..^(♯‘𝐹))((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧))
74		dff13 7056	. 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 586	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 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3054 ↦ cmpt 5124 dom cdm 5540 Fn wfn 6364 ⟶wf 6365 –1-1→wf1 6366 ‘cfv 6369 (class class class)co 7202 ℂcc 10710 0cc0 10712 1c1 10713 + caddc 10715 ℕ₀cn0 12073 ...cfz 13078 ..^cfzo 13221 ♯chash 13879

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-fzo 13222 df-hash 13880

This theorem is referenced by: (None)

W3C validator