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Theorem fargshiftf1 26165

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 6014	. . 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 26164	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → 𝐺:(0..^(#‘𝐹))⟶dom 𝐸)
4	1, 3	sylan2 490	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → 𝐺:(0..^(#‘𝐹))⟶dom 𝐸)
5		ffn 5958	. . . . 5 ⊢ (𝐹:(1...𝑁)⟶dom 𝐸 → 𝐹 Fn (1...𝑁))
6		fseq1hash 13026	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹 Fn (1...𝑁)) → (#‘𝐹) = 𝑁)
7	5, 6	sylan2 490	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (#‘𝐹) = 𝑁)
8	1, 7	sylan2 490	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → (#‘𝐹) = 𝑁)
9		eleq1 2676	. . . . 5 ⊢ ((#‘𝐹) = 𝑁 → ((#‘𝐹) ∈ ℕ₀ ↔ 𝑁 ∈ ℕ₀))
10		oveq2 6557	. . . . . 6 ⊢ ((#‘𝐹) = 𝑁 → (1...(#‘𝐹)) = (1...𝑁))
11		f1eq2 6010	. . . . . 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 743	. . . 4 ⊢ ((#‘𝐹) = 𝑁 → (((#‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(#‘𝐹))–1-1→dom 𝐸) ↔ (𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸)))
14		dff13 6416	. . . . . 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		fargshiftlem 26162	. . . . . . . . . . 11 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (𝑦 + 1) ∈ (1...(#‘𝐹)))
16		fargshiftlem 26162	. . . . . . . . . . 11 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝑧 ∈ (0..^(#‘𝐹))) → (𝑧 + 1) ∈ (1...(#‘𝐹)))
17	15, 16	anim12dan 878	. . . . . . . . . 10 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → ((𝑦 + 1) ∈ (1...(#‘𝐹)) ∧ (𝑧 + 1) ∈ (1...(#‘𝐹))))
18		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑦 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑦 + 1)))
19	18	eqeq1d 2612	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑦 + 1) → ((𝐹‘𝑘) = (𝐹‘𝑙) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘𝑙)))
20		eqeq1 2614	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑦 + 1) → (𝑘 = 𝑙 ↔ (𝑦 + 1) = 𝑙))
21	19, 20	imbi12d 333	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑦 + 1) → (((𝐹‘𝑘) = (𝐹‘𝑙) → 𝑘 = 𝑙) ↔ ((𝐹‘(𝑦 + 1)) = (𝐹‘𝑙) → (𝑦 + 1) = 𝑙)))
22		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = (𝑧 + 1) → (𝐹‘𝑙) = (𝐹‘(𝑧 + 1)))
23	22	eqeq2d 2620	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = (𝑧 + 1) → ((𝐹‘(𝑦 + 1)) = (𝐹‘𝑙) ↔ (𝐹‘(𝑦 + 1)) = (𝐹‘(𝑧 + 1))))
24		eqeq2 2621	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = (𝑧 + 1) → ((𝑦 + 1) = 𝑙 ↔ (𝑦 + 1) = (𝑧 + 1)))
25	23, 24	imbi12d 333	. . . . . . . . . . . . . . . . 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 3293	. . . . . . . . . . . . . . . 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 26163	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) → (𝑦 ∈ (0..^(#‘𝐹)) → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1))))
29	28	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → ((#‘𝐹) ∈ ℕ₀ → (𝑦 ∈ (0..^(#‘𝐹)) → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))))
30	29	com13 86	. . . . . . . . . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1))))
33	32	impcom 445	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) → (𝐺‘𝑦) = (𝐹‘(𝑦 + 1)))
34	2	fargshiftfv 26163	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ 𝐹:(1...(#‘𝐹))⟶dom 𝐸) → (𝑧 ∈ (0..^(#‘𝐹)) → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1))))
35	34	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → ((#‘𝐹) ∈ ℕ₀ → (𝑧 ∈ (0..^(#‘𝐹)) → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))))
36	35	com13 86	. . . . . . . . . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → (𝐹:(1...(#‘𝐹))⟶dom 𝐸 → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1))))
39	38	impcom 445	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:(1...(#‘𝐹))⟶dom 𝐸 ∧ ((#‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹))))) → (𝐺‘𝑧) = (𝐹‘(𝑧 + 1)))
40	33, 39	eqeq12d 2625	. . . . . . . . . . . . . . . . . . 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 12339	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(#‘𝐹)) → 𝑦 ∈ ℤ)
44	43	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . . . . 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 12339	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ (0..^(#‘𝐹)) → 𝑧 ∈ ℤ)
48	47	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . . . . 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 9935	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ (𝑦 ∈ (0..^(#‘𝐹)) ∧ 𝑧 ∈ (0..^(#‘𝐹)))) → 1 ∈ ℂ)
52	46, 50, 51	3jca 1235	. . . . . . . . . . . . . . . . . . . . . 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 10100	. . . . . . . . . . . . . . . . . . . 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 329	. . . . . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . 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 449	. . . . . . . . . . . . . . 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 46	. . . . . . . . . . . . . 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 628	. . . . . . . . . . . . 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 93	. . . . . . . . . . . 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 444	. . . . . . . . . . 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 86	. . . . . . . . . 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 450	. . . . . . . 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 86	. . . . . . 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 2956	. . . . . 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 206	. . . . 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 445	. . . 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 243	. . 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 37	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(1...𝑁)–1-1→dom 𝐸) → ∀𝑦 ∈ (0..^(#‘𝐹))∀𝑧 ∈ (0..^(#‘𝐹))((𝐺‘𝑦) = (𝐺‘𝑧) → 𝑦 = 𝑧))
74		dff13 6416	. 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 695	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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ↦ cmpt 4643 dom cdm 5038 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 ℕ₀cn0 11169 ...cfz 12197 ..^cfzo 12334 #chash 12979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980

This theorem is referenced by: eupatrl 26495

W3C validator