swrdf1 - Mathbox for Thierry Arnoux

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Theorem swrdf1 32590

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 14598	. . . 4 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 substr ⟨𝑀, 𝑁⟩):(0..^(𝑁 − 𝑀))⟶𝐷)
5	1, 2, 3, 4	syl3anc 1368	. . 3 ⊢ (𝜑 → (𝑊 substr ⟨𝑀, 𝑁⟩):(0..^(𝑁 − 𝑀))⟶𝐷)
6	5	ffdmd 6739	. 2 ⊢ (𝜑 → (𝑊 substr ⟨𝑀, 𝑁⟩):dom (𝑊 substr ⟨𝑀, 𝑁⟩)⟶𝐷)
7		fzossz 13650	. . . . . . . 8 ⊢ (0..^(𝑁 − 𝑀)) ⊆ ℤ
8		simpllr 773	. . . . . . . . 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 6719	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑊 substr ⟨𝑀, 𝑁⟩) = (0..^(𝑁 − 𝑀)))
10	9	ad3antrrr 727	. . . . . . . . 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 2827	. . . . . . . 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 3973	. . . . . . 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 12665	. . . . . 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 766	. . . . . . . . 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 2827	. . . . . . . 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 3973	. . . . . . 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 12665	. . . . . 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 13500	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
19	18	ad3antrrr 727	. . . . . . 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 12665	. . . . . 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 727	. . . . . . 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 13495	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ (ℤ_≥‘0))
24		fzoss1 13657	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (ℤ_≥‘0) → (𝑀..^𝑁) ⊆ (0..^𝑁))
25	2, 23, 24	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀..^𝑁) ⊆ (0..^𝑁))
26		elfzuz3 13496	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (0...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ_≥‘𝑁))
27		fzoss2 13658	. . . . . . . . . . . 12 ⊢ ((♯‘𝑊) ∈ (ℤ_≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝑊)))
28	3, 26, 27	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝑊)))
29	25, 28	sstrd 3985	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀..^𝑁) ⊆ (0..^(♯‘𝑊)))
30	29	ad3antrrr 727	. . . . . . . . 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 13500	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
32	31	ad3antrrr 727	. . . . . . . . . 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 13686	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0..^(𝑁 − 𝑀)) ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑖 + 𝑀) ∈ (𝑀..^𝑁))
34	11, 32, 19, 33	syl3anc 1368	. . . . . . . . 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 3976	. . . . . . . 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 14469	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
37	1, 36	syl 17	. . . . . . . . 9 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
38	37	ad3antrrr 727	. . . . . . . 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 2828	. . . . . . 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 13686	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (0..^(𝑁 − 𝑀)) ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 + 𝑀) ∈ (𝑀..^𝑁))
41	15, 32, 19, 40	syl3anc 1368	. . . . . . . . 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 3976	. . . . . . . 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 2828	. . . . . . 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 727	. . . . . . . . 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 727	. . . . . . . . 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 727	. . . . . . . . 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 14596	. . . . . . . . 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 1370	. . . . . . . 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 14596	. . . . . . . . 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 1370	. . . . . . . 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 2772	. . . . . . 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 7249	. . . . . . . . 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 837	. . . . . 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 11418	. . . . 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 3192	. 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 7247	. 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 582	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 1533 ∈ wcel 2098 ∀wral 3053 ⊆ wss 3941 ⟨cop 4627 dom cdm 5667 ⟶wf 6530 –1-1→wf1 6531 ‘cfv 6534 (class class class)co 7402 0cc0 11107 + caddc 11110 − cmin 11442 ℤcz 12556 ℤ_≥cuz 12820 ...cfz 13482 ..^cfzo 13625 ♯chash 14288 Word cword 14462 substr csubstr 14588

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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 df-fzo 13626 df-hash 14289 df-word 14463 df-substr 14589

This theorem is referenced by: cycpmco2f1 32754

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