fargshiftf1 - Mathbox for Alexander van der Vekens

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Theorem fargshiftf1 48728

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)⟶1-1dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶1-1dom 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 6800	. . 3 ⊢ (𝐹:(1...𝑁)⟶1-1dom 𝐸 → 𝐹:(1...𝑁)⟶dom 𝐸)
2		fargshift.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ (0..^(♯‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
3	2	fargshiftf 48727	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
4	1, 3	sylan2 605	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶1-1dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
5		ffn 6731	. . . . 5 ⊢ (𝐹:(1...𝑁)⟶dom 𝐸 → 𝐹 Fn (1...𝑁))
6		fseq1hash 14547	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)
7	5, 6	sylan2 605	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (♯‘𝐹) = 𝑁)
8	1, 7	sylan2 605	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶1-1dom 𝐸) → (♯‘𝐹) = 𝑁)
9		eleq1 2852	. . . . 5 ⊢ ((♯‘𝐹) = 𝑁 → ((♯‘𝐹) ∈ ℕ₀ ↔ 𝑁 ∈ ℕ₀))
10		oveq2 7451	. . . . . 6 ⊢ ((♯‘𝐹) = 𝑁 → (1...(♯‘𝐹)) = (1...𝑁))
11		f1eq2 6796	. . . . . 6 ⊢ ((1...(♯‘𝐹)) = (1...𝑁) → (𝐹:(1...(♯‘𝐹))⟶1-1dom 𝐸 ↔ 𝐹:(1...𝑁)⟶1-1dom 𝐸))
12	10, 11	syl 18	. . . . 5 ⊢ ((♯‘𝐹) = 𝑁 → (𝐹:(1...(♯‘𝐹))⟶1-1dom 𝐸 ↔ 𝐹:(1...𝑁)⟶1-1dom 𝐸))
13	9, 12	anbi12d 644	. . . 4 ⊢ ((♯‘𝐹) = 𝑁 → (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶1-1dom 𝐸) ↔ (𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶1-1dom 𝐸)))
14		dff13 7281	. . . . . 6 ⊢ (𝐹:(1...(♯‘𝐹))⟶1-1dom 𝐸 ↔ (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ∀𝑘 ∈ (1...(♯‘𝐹))∀𝑙 ∈ (1...(♯‘𝐹))((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙)))
15		fz0add1fz1 13897	. . . . . . . . . . 11 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (𝑦 + 1) ∈ (1...(♯‘𝐹)))
16		fz0add1fz1 13897	. . . . . . . . . . 11 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → (𝑧 + 1) ∈ (1...(♯‘𝐹)))
17	15, 16	anim12dan 631	. . . . . . . . . 10 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → ((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹))))
18		fveq2 6908	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑦 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑦 + 1)))
19	18	eqeq1d 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑦 + 1) → ((𝐹‘𝑘) = (𝐹‘𝑙) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘𝑙)))
20		eqeq1 2767	. . . . . . . . . . . . . . . . . 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 6908	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = (𝑧 + 1) → (𝐹‘𝑙) = (𝐹‘(𝑧 + 1)))
23	22	eqeq2d 2774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = (𝑧 + 1) → ((𝐹‘(𝑦 + 1)) = (𝐹‘𝑙) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1))))
24		eqeq2 2775	. . . . . . . . . . . . . . . . . 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 3590	. . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . 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 48726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → (𝑦 ∈ (0..^(♯‘𝐹)) → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1))))
29	28	expcom 419	. . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → ((♯‘𝐹) ∈ ℕ₀ → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))))
32	31	impcom 413	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1))))
33	32	impcom 413	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))
34	2	fargshiftfv 48726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → (𝑧 ∈ (0..^(♯‘𝐹)) → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1))))
35	34	expcom 419	. . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → ((♯‘𝐹) ∈ ℕ₀ → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))))
38	37	impcom 413	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1))))
39	38	impcom 413	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))
40	33, 39	eqeq12d 2779	. . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 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 13820	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ ℤ)
44	43	zcnd 12831	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ ℂ)
45	44	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → 𝑦 ∈ ℂ)
46	45	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → 𝑦 ∈ ℂ)
47		elfzoelz 13820	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ (0..^(♯‘𝐹)) → 𝑧 ∈ ℤ)
48	47	zcnd 12831	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∈ (0..^(♯‘𝐹)) → 𝑧 ∈ ℂ)
49	48	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → 𝑧 ∈ ℂ)
50	49	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → 𝑧 ∈ ℂ)
51		1cnd 11329	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → 1 ∈ ℂ)
52	46, 50, 51	3jca 1146	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ))
53	52	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ))
54	53	adantr 486	. . . . . . . . . . . . . . . . . . . 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 11522	. . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . 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 425	. . . . . . . . . . . . 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 412	. . . . . . . . . . 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 419	. . . . . . . 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 3210	. . . . . 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))⟶1-1dom 𝐸 → ((♯‘𝐹) ∈ ℕ₀ → ∀𝑦 ∈ (0..^(♯‘𝐹))∀𝑧 ∈ (0..^(♯‘𝐹))((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧)))
71	70	impcom 413	. . . 4 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶1-1dom 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)⟶1-1dom 𝐸) → ∀𝑦 ∈ (0..^(♯‘𝐹))∀𝑧 ∈ (0..^(♯‘𝐹))((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧)))
73	8, 72	mpcom 39	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶1-1dom 𝐸) → ∀𝑦 ∈ (0..^(♯‘𝐹))∀𝑧 ∈ (0..^(♯‘𝐹))((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧))
74		dff13 7281	. 2 ⊢ (𝐺:(0..^(♯‘𝐹))⟶1-1dom 𝐸 ↔ (𝐺:(0..^(♯‘𝐹))⟶dom 𝐸 ∧ ∀𝑦 ∈ (0..^(♯‘𝐹))∀𝑧 ∈ (0..^(♯‘𝐹))((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧)))
75	4, 73, 74	sylanbrc 595	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶1-1dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶1-1dom 𝐸)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ↦ cmpt 5189 dom cdm 5655 Fn wfn 6553 ⟶wf 6554 ⟶1-1wf1 6555 ‘cfv 6558 (class class class)co 7443 ℂcc 11225 0cc0 11227 1c1 11228 + caddc 11230 ℕ₀cn0 12633 ...cfz 13666 ..^cfzo 13815 ♯chash 14501

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-er 8745 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-n0 12634 df-z 12721 df-uz 12993 df-fz 13667 df-fzo 13816 df-hash 14502

This theorem is used by: (None)

W3C validator