swrdf1 - Mathbox for Thierry Arnoux

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Theorem swrdf1 32115

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 14599	. . . 4 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 substr ⟨𝑀, 𝑁⟩):(0..^(𝑁 − 𝑀))⟶𝐷)
5	1, 2, 3, 4	syl3anc 1371	. . 3 ⊢ (𝜑 → (𝑊 substr ⟨𝑀, 𝑁⟩):(0..^(𝑁 − 𝑀))⟶𝐷)
6	5	ffdmd 6748	. 2 ⊢ (𝜑 → (𝑊 substr ⟨𝑀, 𝑁⟩):dom (𝑊 substr ⟨𝑀, 𝑁⟩)⟶𝐷)
7		fzossz 13651	. . . . . . . 8 ⊢ (0..^(𝑁 − 𝑀)) ⊆ ℤ
8		simpllr 774	. . . . . . . . 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 6728	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑊 substr ⟨𝑀, 𝑁⟩) = (0..^(𝑁 − 𝑀)))
10	9	ad3antrrr 728	. . . . . . . . 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 2835	. . . . . . . 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 3980	. . . . . . 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 12666	. . . . . 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 767	. . . . . . . . 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 2835	. . . . . . . 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 3980	. . . . . . 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 12666	. . . . . 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 13501	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
19	18	ad3antrrr 728	. . . . . . 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 12666	. . . . . 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 728	. . . . . . 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 13496	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ (ℤ_≥‘0))
24		fzoss1 13658	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (ℤ_≥‘0) → (𝑀..^𝑁) ⊆ (0..^𝑁))
25	2, 23, 24	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀..^𝑁) ⊆ (0..^𝑁))
26		elfzuz3 13497	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ_≥‘𝑁))
27		fzoss2 13659	. . . . . . . . . . . 12 ⊢ ((♯‘𝑊) ∈ (ℤ_≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝑊)))
28	3, 26, 27	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝑊)))
29	25, 28	sstrd 3992	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀..^𝑁) ⊆ (0..^(♯‘𝑊)))
30	29	ad3antrrr 728	. . . . . . . . 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 13501	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
32	31	ad3antrrr 728	. . . . . . . . . 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 13687	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0..^(𝑁 − 𝑀)) ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑖 + 𝑀) ∈ (𝑀..^𝑁))
34	11, 32, 19, 33	syl3anc 1371	. . . . . . . . 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 3983	. . . . . . . 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 14470	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
37	1, 36	syl 17	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
38	37	ad3antrrr 728	. . . . . . . 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 2836	. . . . . . 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 13687	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (0..^(𝑁 − 𝑀)) ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 + 𝑀) ∈ (𝑀..^𝑁))
41	15, 32, 19, 40	syl3anc 1371	. . . . . . . . 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 3983	. . . . . . . 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 2836	. . . . . . 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 485	. . . . . . . 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 728	. . . . . . . . 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 728	. . . . . . . . 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 728	. . . . . . . . 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 14597	. . . . . . . . 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 1373	. . . . . . . 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 14597	. . . . . . . . 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 1373	. . . . . . . 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 2780	. . . . . . 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 7255	. . . . . . . . 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 468	. . . . . . . 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 407	. . . . . . 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 838	. . . . . 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 11419	. . . . 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 413	. . . 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 467	. . 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 3200	. 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 7253	. 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 583	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 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ⊆ wss 3948 ⟨cop 4634 dom cdm 5676 ⟶wf 6539 –1-1→wf1 6540 ‘cfv 6543 (class class class)co 7408 0cc0 11109 + caddc 11112 − cmin 11443 ℤcz 12557 ℤ_≥cuz 12821 ...cfz 13483 ..^cfzo 13626 ♯chash 14289 Word cword 14463 substr csubstr 14589

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-fzo 13627 df-hash 14290 df-word 14464 df-substr 14590

This theorem is referenced by: cycpmco2f1 32278

W3C validator