swrdf1 - Mathbox for Thierry Arnoux

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Theorem swrdf1 33016

Description: Condition for a subword to be injective. (Contributed by Thierry Arnoux, 12-Dec-2023.)

**Hypotheses**
Ref	Expression
swrdf1.w	⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
swrdf1.m	⊢ (𝜑 → 𝑀 ∈ (0...𝑁))
swrdf1.n	⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝑊)))
swrdf1.1	⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)

**Assertion**
Ref	Expression
swrdf1	⊢ (𝜑 → (𝑊 substr ⟨𝑀, 𝑁⟩):dom (𝑊 substr ⟨𝑀, 𝑁⟩)–1-1→𝐷)

Proof of Theorem swrdf1 Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		swrdf1.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
2		swrdf1.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (0...𝑁))
3		swrdf1.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝑊)))
4		swrdf 14613	. . . 4 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 substr ⟨𝑀, 𝑁⟩):(0..^(𝑁 − 𝑀))⟶𝐷)
5	1, 2, 3, 4	syl3anc 1374	. . 3 ⊢ (𝜑 → (𝑊 substr ⟨𝑀, 𝑁⟩):(0..^(𝑁 − 𝑀))⟶𝐷)
6	5	ffdmd 6698	. 2 ⊢ (𝜑 → (𝑊 substr ⟨𝑀, 𝑁⟩):dom (𝑊 substr ⟨𝑀, 𝑁⟩)⟶𝐷)
7		fzossz 13634	. . . . . . . 8 ⊢ (0..^(𝑁 − 𝑀)) ⊆ ℤ
8		simpllr 776	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩))
9	5	fdmd 6678	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑊 substr ⟨𝑀, 𝑁⟩) = (0..^(𝑁 − 𝑀)))
10	9	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → dom (𝑊 substr ⟨𝑀, 𝑁⟩) = (0..^(𝑁 − 𝑀)))
11	8, 10	eleqtrd 2838	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑖 ∈ (0..^(𝑁 − 𝑀)))
12	7, 11	sselid 3919	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑖 ∈ ℤ)
13	12	zcnd 12634	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑖 ∈ ℂ)
14		simplr 769	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩))
15	14, 10	eleqtrd 2838	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑗 ∈ (0..^(𝑁 − 𝑀)))
16	7, 15	sselid 3919	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑗 ∈ ℤ)
17	16	zcnd 12634	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑗 ∈ ℂ)
18	2	elfzelzd 13479	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
19	18	ad3antrrr 731	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑀 ∈ ℤ)
20	19	zcnd 12634	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑀 ∈ ℂ)
21		swrdf1.1	. . . . . . . 8 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
22	21	ad3antrrr 731	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑊:dom 𝑊–1-1→𝐷)
23		elfzuz 13474	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ (ℤ_≥‘0))
24		fzoss1 13641	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (ℤ_≥‘0) → (𝑀..^𝑁) ⊆ (0..^𝑁))
25	2, 23, 24	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀..^𝑁) ⊆ (0..^𝑁))
26		elfzuz3 13475	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ_≥‘𝑁))
27		fzoss2 13642	. . . . . . . . . . . 12 ⊢ ((♯‘𝑊) ∈ (ℤ_≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝑊)))
28	3, 26, 27	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝑊)))
29	25, 28	sstrd 3932	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀..^𝑁) ⊆ (0..^(♯‘𝑊)))
30	29	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → (𝑀..^𝑁) ⊆ (0..^(♯‘𝑊)))
31	3	elfzelzd 13479	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
32	31	ad3antrrr 731	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑁 ∈ ℤ)
33		fzoaddel2 13675	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0..^(𝑁 − 𝑀)) ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑖 + 𝑀) ∈ (𝑀..^𝑁))
34	11, 32, 19, 33	syl3anc 1374	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → (𝑖 + 𝑀) ∈ (𝑀..^𝑁))
35	30, 34	sseldd 3922	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → (𝑖 + 𝑀) ∈ (0..^(♯‘𝑊)))
36		wrddm 14483	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
37	1, 36	syl 17	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
38	37	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → dom 𝑊 = (0..^(♯‘𝑊)))
39	35, 38	eleqtrrd 2839	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → (𝑖 + 𝑀) ∈ dom 𝑊)
40		fzoaddel2 13675	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (0..^(𝑁 − 𝑀)) ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 + 𝑀) ∈ (𝑀..^𝑁))
41	15, 32, 19, 40	syl3anc 1374	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → (𝑗 + 𝑀) ∈ (𝑀..^𝑁))
42	30, 41	sseldd 3922	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → (𝑗 + 𝑀) ∈ (0..^(♯‘𝑊)))
43	42, 38	eleqtrrd 2839	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → (𝑗 + 𝑀) ∈ dom 𝑊)
44		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗))
45	1	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑊 ∈ Word 𝐷)
46	2	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑀 ∈ (0...𝑁))
47	3	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑁 ∈ (0...(♯‘𝑊)))
48		swrdfv 14611	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝐷 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑖 ∈ (0..^(𝑁 − 𝑀))) → ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = (𝑊‘(𝑖 + 𝑀)))
49	45, 46, 47, 11, 48	syl31anc 1376	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = (𝑊‘(𝑖 + 𝑀)))
50		swrdfv 14611	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝐷 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝑗 ∈ (0..^(𝑁 − 𝑀))) → ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = (𝑊‘(𝑗 + 𝑀)))
51	45, 46, 47, 15, 50	syl31anc 1376	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = (𝑊‘(𝑗 + 𝑀)))
52	44, 49, 51	3eqtr3d 2779	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → (𝑊‘(𝑖 + 𝑀)) = (𝑊‘(𝑗 + 𝑀)))
53		f1veqaeq 7211	. . . . . . . . 9 ⊢ ((𝑊:dom 𝑊–1-1→𝐷 ∧ ((𝑖 + 𝑀) ∈ dom 𝑊 ∧ (𝑗 + 𝑀) ∈ dom 𝑊)) → ((𝑊‘(𝑖 + 𝑀)) = (𝑊‘(𝑗 + 𝑀)) → (𝑖 + 𝑀) = (𝑗 + 𝑀)))
54	53	anassrs 467	. . . . . . . 8 ⊢ (((𝑊:dom 𝑊–1-1→𝐷 ∧ (𝑖 + 𝑀) ∈ dom 𝑊) ∧ (𝑗 + 𝑀) ∈ dom 𝑊) → ((𝑊‘(𝑖 + 𝑀)) = (𝑊‘(𝑗 + 𝑀)) → (𝑖 + 𝑀) = (𝑗 + 𝑀)))
55	54	imp 406	. . . . . . 7 ⊢ ((((𝑊:dom 𝑊–1-1→𝐷 ∧ (𝑖 + 𝑀) ∈ dom 𝑊) ∧ (𝑗 + 𝑀) ∈ dom 𝑊) ∧ (𝑊‘(𝑖 + 𝑀)) = (𝑊‘(𝑗 + 𝑀))) → (𝑖 + 𝑀) = (𝑗 + 𝑀))
56	22, 39, 43, 52, 55	syl1111anc 841	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → (𝑖 + 𝑀) = (𝑗 + 𝑀))
57	13, 17, 20, 56	addcan2ad 11352	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗)) → 𝑖 = 𝑗)
58	57	ex 412	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)) → (((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) → 𝑖 = 𝑗))
59	58	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩) ∧ 𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩))) → (((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) → 𝑖 = 𝑗))
60	59	ralrimivva 3180	. 2 ⊢ (𝜑 → ∀𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)∀𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)(((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) → 𝑖 = 𝑗))
61		dff13 7209	. 2 ⊢ ((𝑊 substr ⟨𝑀, 𝑁⟩):dom (𝑊 substr ⟨𝑀, 𝑁⟩)–1-1→𝐷 ↔ ((𝑊 substr ⟨𝑀, 𝑁⟩):dom (𝑊 substr ⟨𝑀, 𝑁⟩)⟶𝐷 ∧ ∀𝑖 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)∀𝑗 ∈ dom (𝑊 substr ⟨𝑀, 𝑁⟩)(((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) → 𝑖 = 𝑗)))
62	6, 60, 61	sylanbrc 584	1 ⊢ (𝜑 → (𝑊 substr ⟨𝑀, 𝑁⟩):dom (𝑊 substr ⟨𝑀, 𝑁⟩)–1-1→𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3051 ⊆ wss 3889 ⟨cop 4573 dom cdm 5631 ⟶wf 6494 –1-1→wf1 6495 ‘cfv 6498 (class class class)co 7367 0cc0 11038 + caddc 11041 − cmin 11377 ℤcz 12524 ℤ_≥cuz 12788 ...cfz 13461 ..^cfzo 13608 ♯chash 14292 Word cword 14475 substr csubstr 14603

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-fzo 13609 df-hash 14293 df-word 14476 df-substr 14604

This theorem is referenced by: cycpmco2f1 33185

W3C validator