fargshiftf1 - Mathbox for Alexander van der Vekens

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Theorem fargshiftf1 47446

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 6759	. . 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 47445	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
4	1, 3	sylan2 593	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
5		ffn 6691	. . . . 5 ⊢ (𝐹:(1...𝑁)⟶dom 𝐸 → 𝐹 Fn (1...𝑁))
6		fseq1hash 14348	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)
7	5, 6	sylan2 593	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (♯‘𝐹) = 𝑁)
8	1, 7	sylan2 593	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → (♯‘𝐹) = 𝑁)
9		eleq1 2817	. . . . 5 ⊢ ((♯‘𝐹) = 𝑁 → ((♯‘𝐹) ∈ ℕ₀ ↔ 𝑁 ∈ ℕ₀))
10		oveq2 7398	. . . . . 6 ⊢ ((♯‘𝐹) = 𝑁 → (1...(♯‘𝐹)) = (1...𝑁))
11		f1eq2 6755	. . . . . 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 632	. . . 4 ⊢ ((♯‘𝐹) = 𝑁 → (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))–1-1→dom 𝐸) ↔ (𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸)))
14		dff13 7232	. . . . . 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 13703	. . . . . . . . . . 11 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (𝑦 + 1) ∈ (1...(♯‘𝐹)))
16		fz0add1fz1 13703	. . . . . . . . . . 11 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → (𝑧 + 1) ∈ (1...(♯‘𝐹)))
17	15, 16	anim12dan 619	. . . . . . . . . 10 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → ((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹))))
18		fveq2 6861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑦 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑦 + 1)))
19	18	eqeq1d 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑦 + 1) → ((𝐹‘𝑘) = (𝐹‘𝑙) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘𝑙)))
20		eqeq1 2734	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑦 + 1) → (𝑘 = 𝑙 ↔ (𝑦 + 1) = 𝑙))
21	19, 20	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑦 + 1) → (((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙) ↔ ((𝐹‘(𝑦 + 1)) = (𝐹‘𝑙) → (𝑦 + 1) = 𝑙)))
22		fveq2 6861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = (𝑧 + 1) → (𝐹‘𝑙) = (𝐹‘(𝑧 + 1)))
23	22	eqeq2d 2741	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = (𝑧 + 1) → ((𝐹‘(𝑦 + 1)) = (𝐹‘𝑙) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1))))
24		eqeq2 2742	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = (𝑧 + 1) → ((𝑦 + 1) = 𝑙 ↔ (𝑦 + 1) = (𝑧 + 1)))
25	23, 24	imbi12d 344	. . . . . . . . . . . . . . . . 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 3602	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 47444	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → (𝑦 ∈ (0..^(♯‘𝐹)) → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1))))
29	28	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → ((♯‘𝐹) ∈ ℕ₀ → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))))
32	31	impcom 407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1))))
33	32	impcom 407	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))
34	2	fargshiftfv 47444	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → (𝑧 ∈ (0..^(♯‘𝐹)) → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1))))
35	34	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → ((♯‘𝐹) ∈ ℕ₀ → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))))
38	37	impcom 407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1))))
39	38	impcom 407	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))
40	33, 39	eqeq12d 2746	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 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 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))) → ((𝐺‘𝑦) = (𝐺‘𝑧) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1))))
43		elfzoelz 13627	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ ℤ)
44	43	zcnd 12646	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ ℂ)
45	44	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → 𝑦 ∈ ℂ)
46	45	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → 𝑦 ∈ ℂ)
47		elfzoelz 13627	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ (0..^(♯‘𝐹)) → 𝑧 ∈ ℤ)
48	47	zcnd 12646	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∈ (0..^(♯‘𝐹)) → 𝑧 ∈ ℂ)
49	48	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → 𝑧 ∈ ℂ)
50	49	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → 𝑧 ∈ ℂ)
51		1cnd 11176	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → 1 ∈ ℂ)
52	46, 50, 51	3jca 1128	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ))
53	52	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ))
54	53	adantr 480	. . . . . . . . . . . . . . . . . . . 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 11366	. . . . . . . . . . . . . . . . . . . 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 340	. . . . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . 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 406	. . . . . . . . . . 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 413	. . . . . . . 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 3182	. . . . . 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 217	. . . . 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 407	. . . 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 254	. . 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 7232	. 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 583	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ↦ cmpt 5191 dom cdm 5641 Fn wfn 6509 ⟶wf 6510 –1-1→wf1 6511 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 0cc0 11075 1c1 11076 + caddc 11078 ℕ₀cn0 12449 ...cfz 13475 ..^cfzo 13622 ♯chash 14302

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-fzo 13623 df-hash 14303

This theorem is referenced by: (None)

W3C validator