fargshiftf1 - Mathbox for Alexander van der Vekens

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Theorem fargshiftf1 46844

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 6791	. . 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 46843	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
4	1, 3	sylan2 591	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(♯‘𝐹))⟶dom 𝐸)
5		ffn 6721	. . . . 5 ⊢ (𝐹:(1...𝑁)⟶dom 𝐸 → 𝐹 Fn (1...𝑁))
6		fseq1hash 14367	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹 Fn (1...𝑁)) → (♯‘𝐹) = 𝑁)
7	5, 6	sylan2 591	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (♯‘𝐹) = 𝑁)
8	1, 7	sylan2 591	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → (♯‘𝐹) = 𝑁)
9		eleq1 2813	. . . . 5 ⊢ ((♯‘𝐹) = 𝑁 → ((♯‘𝐹) ∈ ℕ₀ ↔ 𝑁 ∈ ℕ₀))
10		oveq2 7425	. . . . . 6 ⊢ ((♯‘𝐹) = 𝑁 → (1...(♯‘𝐹)) = (1...𝑁))
11		f1eq2 6787	. . . . . 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 630	. . . 4 ⊢ ((♯‘𝐹) = 𝑁 → (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))–1-1→dom 𝐸) ↔ (𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸)))
14		dff13 7263	. . . . . 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 13734	. . . . . . . . . . 11 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑦 ∈ (0..^(♯‘𝐹))) → (𝑦 + 1) ∈ (1...(♯‘𝐹)))
16		fz0add1fz1 13734	. . . . . . . . . . 11 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → (𝑧 + 1) ∈ (1...(♯‘𝐹)))
17	15, 16	anim12dan 617	. . . . . . . . . 10 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → ((𝑦 + 1) ∈ (1...(♯‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(♯‘𝐹))))
18		fveq2 6894	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑦 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑦 + 1)))
19	18	eqeq1d 2727	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑦 + 1) → ((𝐹‘𝑘) = (𝐹‘𝑙) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘𝑙)))
20		eqeq1 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑦 + 1) → (𝑘 = 𝑙 ↔ (𝑦 + 1) = 𝑙))
21	19, 20	imbi12d 343	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑦 + 1) → (((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙) ↔ ((𝐹‘(𝑦 + 1)) = (𝐹‘𝑙) → (𝑦 + 1) = 𝑙)))
22		fveq2 6894	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = (𝑧 + 1) → (𝐹‘𝑙) = (𝐹‘(𝑧 + 1)))
23	22	eqeq2d 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = (𝑧 + 1) → ((𝐹‘(𝑦 + 1)) = (𝐹‘𝑙) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1))))
24		eqeq2 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = (𝑧 + 1) → ((𝑦 + 1) = 𝑙 ↔ (𝑦 + 1) = (𝑧 + 1)))
25	23, 24	imbi12d 343	. . . . . . . . . . . . . . . . 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 3618	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 46842	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → (𝑦 ∈ (0..^(♯‘𝐹)) → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1))))
29	28	expcom 412	. . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → ((♯‘𝐹) ∈ ℕ₀ → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))))
32	31	impcom 406	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1))))
33	32	impcom 406	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))
34	2	fargshiftfv 46842	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(♯‘𝐹))⟶dom 𝐸) → (𝑧 ∈ (0..^(♯‘𝐹)) → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1))))
35	34	expcom 412	. . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → ((♯‘𝐹) ∈ ℕ₀ → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))))
38	37	impcom 406	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (𝐹:(1...(♯‘𝐹))⟶dom 𝐸 → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1))))
39	38	impcom 406	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))
40	33, 39	eqeq12d 2741	. . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . 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 13664	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ ℤ)
44	43	zcnd 12697	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (0..^(♯‘𝐹)) → 𝑦 ∈ ℂ)
45	44	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → 𝑦 ∈ ℂ)
46	45	adantl 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → 𝑦 ∈ ℂ)
47		elfzoelz 13664	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ (0..^(♯‘𝐹)) → 𝑧 ∈ ℤ)
48	47	zcnd 12697	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∈ (0..^(♯‘𝐹)) → 𝑧 ∈ ℂ)
49	48	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))) → 𝑧 ∈ ℂ)
50	49	adantl 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → 𝑧 ∈ ℂ)
51		1cnd 11239	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → 1 ∈ ℂ)
52	46, 50, 51	3jca 1125	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹)))) → (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ))
53	52	adantl 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(1...(♯‘𝐹))⟶dom 𝐸 ∧ ((♯‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(♯‘𝐹)) ∧ 𝑧 ∈ (0..^(♯‘𝐹))))) → (𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 1 ∈ ℂ))
54	53	adantr 479	. . . . . . . . . . . . . . . . . . . 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 11429	. . . . . . . . . . . . . . . . . . . 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 339	. . . . . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . . . . 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 239	. . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . 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 405	. . . . . . . . . . 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 412	. . . . . . . 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 3192	. . . . . 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 216	. . . . 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 406	. . . 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 253	. . 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 7263	. 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 581	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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ↦ cmpt 5231 dom cdm 5677 Fn wfn 6542 ⟶wf 6543 –1-1→wf1 6544 ‘cfv 6547 (class class class)co 7417 ℂcc 11136 0cc0 11138 1c1 11139 + caddc 11141 ℕ₀cn0 12502 ...cfz 13516 ..^cfzo 13659 ♯chash 14321

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7870 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-hash 14322

This theorem is referenced by: (None)

W3C validator