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Theorem swrdwrdsymbg 11458

Description: A subword is a word over the symbols it consists of. (Contributed by AV, 2-Dec-2022.)

**Assertion**
Ref	Expression
swrdwrdsymbg	⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆))) → (𝑆 substr ⟨𝑀, 𝑁⟩) ∈ Word (𝑆 “ (𝑀..^𝑁)))

Proof of Theorem swrdwrdsymbg Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		swrdval2 11445	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆))) → (𝑆 substr ⟨𝑀, 𝑁⟩) = (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))))
2	1	3expb 1235	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑀, 𝑁⟩) = (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))))
3		wrdf 11332	. . . . . . . . . 10 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
4	3	ffund 5540	. . . . . . . . 9 ⊢ (𝑆 ∈ Word 𝐴 → Fun 𝑆)
5	4	adantr 276	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → Fun 𝑆)
6	5	adantr 276	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → Fun 𝑆)
7		wrddm 11334	. . . . . . . . 9 ⊢ (𝑆 ∈ Word 𝐴 → dom 𝑆 = (0..^(♯‘𝑆)))
8		elfzodifsumelfzo 10636	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆))) → (𝑥 ∈ (0..^(𝑁 − 𝑀)) → (𝑥 + 𝑀) ∈ (0..^(♯‘𝑆))))
9	8	imp 124	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑥 + 𝑀) ∈ (0..^(♯‘𝑆)))
10	9	adantl 277	. . . . . . . . . . 11 ⊢ ((dom 𝑆 = (0..^(♯‘𝑆)) ∧ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀)))) → (𝑥 + 𝑀) ∈ (0..^(♯‘𝑆)))
11		eleq2 2302	. . . . . . . . . . . 12 ⊢ (dom 𝑆 = (0..^(♯‘𝑆)) → ((𝑥 + 𝑀) ∈ dom 𝑆 ↔ (𝑥 + 𝑀) ∈ (0..^(♯‘𝑆))))
12	11	adantr 276	. . . . . . . . . . 11 ⊢ ((dom 𝑆 = (0..^(♯‘𝑆)) ∧ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀)))) → ((𝑥 + 𝑀) ∈ dom 𝑆 ↔ (𝑥 + 𝑀) ∈ (0..^(♯‘𝑆))))
13	10, 12	mpbird 167	. . . . . . . . . 10 ⊢ ((dom 𝑆 = (0..^(♯‘𝑆)) ∧ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀)))) → (𝑥 + 𝑀) ∈ dom 𝑆)
14	13	exp32 365	. . . . . . . . 9 ⊢ (dom 𝑆 = (0..^(♯‘𝑆)) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆))) → (𝑥 ∈ (0..^(𝑁 − 𝑀)) → (𝑥 + 𝑀) ∈ dom 𝑆)))
15	7, 14	syl 14	. . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐴 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆))) → (𝑥 ∈ (0..^(𝑁 − 𝑀)) → (𝑥 + 𝑀) ∈ dom 𝑆)))
16	15	imp31 256	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑥 + 𝑀) ∈ dom 𝑆)
17		simpr 110	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → 𝑥 ∈ (0..^(𝑁 − 𝑀)))
18		elfzelz 10445	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (0...(♯‘𝑆)) → 𝑁 ∈ ℤ)
19	18	adantl 277	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆))) → 𝑁 ∈ ℤ)
20	19	adantl 277	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → 𝑁 ∈ ℤ)
21	20	adantr 276	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → 𝑁 ∈ ℤ)
22		elfzelz 10445	. . . . . . . . . 10 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℤ)
23	22	ad2antrl 494	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → 𝑀 ∈ ℤ)
24	23	adantr 276	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → 𝑀 ∈ ℤ)
25		fzoaddel2 10625	. . . . . . . 8 ⊢ ((𝑥 ∈ (0..^(𝑁 − 𝑀)) ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑥 + 𝑀) ∈ (𝑀..^𝑁))
26	17, 21, 24, 25	syl3anc 1278	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑥 + 𝑀) ∈ (𝑀..^𝑁))
27		funfvima 5953	. . . . . . . 8 ⊢ ((Fun 𝑆 ∧ (𝑥 + 𝑀) ∈ dom 𝑆) → ((𝑥 + 𝑀) ∈ (𝑀..^𝑁) → (𝑆‘(𝑥 + 𝑀)) ∈ (𝑆 “ (𝑀..^𝑁))))
28	27	imp 124	. . . . . . 7 ⊢ (((Fun 𝑆 ∧ (𝑥 + 𝑀) ∈ dom 𝑆) ∧ (𝑥 + 𝑀) ∈ (𝑀..^𝑁)) → (𝑆‘(𝑥 + 𝑀)) ∈ (𝑆 “ (𝑀..^𝑁)))
29	6, 16, 26, 28	syl21anc 1277	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑆‘(𝑥 + 𝑀)) ∈ (𝑆 “ (𝑀..^𝑁)))
30	29	fmpttd 5866	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))):(0..^(𝑁 − 𝑀))⟶(𝑆 “ (𝑀..^𝑁)))
31		simpll 531	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → 𝑆 ∈ Word 𝐴)
32		elfzoelz 10571	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0..^(𝑁 − 𝑀)) → 𝑥 ∈ ℤ)
33	32	adantl 277	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → 𝑥 ∈ ℤ)
34	33, 24	zaddcld 9783	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑥 + 𝑀) ∈ ℤ)
35		fvexg 5717	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑥 + 𝑀) ∈ ℤ) → (𝑆‘(𝑥 + 𝑀)) ∈ V)
36	31, 34, 35	syl2anc 415	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑆‘(𝑥 + 𝑀)) ∈ V)
37	36	ralrimiva 2623	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → ∀𝑥 ∈ (0..^(𝑁 − 𝑀))(𝑆‘(𝑥 + 𝑀)) ∈ V)
38		eqid 2238	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))) = (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀)))
39	38	fnmpt 5513	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ (0..^(𝑁 − 𝑀))(𝑆‘(𝑥 + 𝑀)) ∈ V → (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))) Fn (0..^(𝑁 − 𝑀)))
40	37, 39	syl 14	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))) Fn (0..^(𝑁 − 𝑀)))
41		0z 9666	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
42		elfzel2 10443	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
43	42, 22	zsubcld 9784	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → (𝑁 − 𝑀) ∈ ℤ)
44		fzofig 10890	. . . . . . . . . . 11 ⊢ ((0 ∈ ℤ ∧ (𝑁 − 𝑀) ∈ ℤ) → (0..^(𝑁 − 𝑀)) ∈ Fin)
45	41, 43, 44	sylancr 418	. . . . . . . . . 10 ⊢ (𝑀 ∈ (0...𝑁) → (0..^(𝑁 − 𝑀)) ∈ Fin)
46	45	ad2antrl 494	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (0..^(𝑁 − 𝑀)) ∈ Fin)
47		fihashfn 11262	. . . . . . . . 9 ⊢ (((𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))) Fn (0..^(𝑁 − 𝑀)) ∧ (0..^(𝑁 − 𝑀)) ∈ Fin) → (♯‘(𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀)))) = (♯‘(0..^(𝑁 − 𝑀))))
48	40, 46, 47	syl2anc 415	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀)))) = (♯‘(0..^(𝑁 − 𝑀))))
49		fznn0sub 10480	. . . . . . . . . 10 ⊢ (𝑀 ∈ (0...𝑁) → (𝑁 − 𝑀) ∈ ℕ₀)
50		hashfzo0 11286	. . . . . . . . . 10 ⊢ ((𝑁 − 𝑀) ∈ ℕ₀ → (♯‘(0..^(𝑁 − 𝑀))) = (𝑁 − 𝑀))
51	49, 50	syl 14	. . . . . . . . 9 ⊢ (𝑀 ∈ (0...𝑁) → (♯‘(0..^(𝑁 − 𝑀))) = (𝑁 − 𝑀))
52	51	ad2antrl 494	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (♯‘(0..^(𝑁 − 𝑀))) = (𝑁 − 𝑀))
53	48, 52	eqtrd 2271	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀)))) = (𝑁 − 𝑀))
54	53	oveq2d 6104	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘(𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))))) = (0..^(𝑁 − 𝑀)))
55	54	feq2d 5524	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → ((𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))):(0..^(♯‘(𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀)))))⟶(𝑆 “ (𝑀..^𝑁)) ↔ (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))):(0..^(𝑁 − 𝑀))⟶(𝑆 “ (𝑀..^𝑁))))
56	30, 55	mpbird 167	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))):(0..^(♯‘(𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀)))))⟶(𝑆 “ (𝑀..^𝑁)))
57	49	ad2antrl 494	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (𝑁 − 𝑀) ∈ ℕ₀)
58	53, 57	eqeltrd 2315	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀)))) ∈ ℕ₀)
59		iswrdinn0 11331	. . . 4 ⊢ (((𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))):(0..^(♯‘(𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀)))))⟶(𝑆 “ (𝑀..^𝑁)) ∧ (♯‘(𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀)))) ∈ ℕ₀) → (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))) ∈ Word (𝑆 “ (𝑀..^𝑁)))
60	56, 58, 59	syl2anc 415	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))) ∈ Word (𝑆 “ (𝑀..^𝑁)))
61	2, 60	eqeltrd 2315	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑀, 𝑁⟩) ∈ Word (𝑆 “ (𝑀..^𝑁)))
62	61	3impb 1230	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆))) → (𝑆 substr ⟨𝑀, 𝑁⟩) ∈ Word (𝑆 “ (𝑀..^𝑁)))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ∀wral 2528 Vcvv 2821 ⟨cop 3712 ↦ cmpt 4193 dom cdm 4776 “ cima 4779 Fun wfun 5374 Fn wfn 5375 ⟶wf 5376 ‘cfv 5380 (class class class)co 6088 Fincfn 7026 0cc0 8185 + caddc 8188 − cmin 8504 ℕ₀cn0 9574 ℤcz 9655 ...cfz 10428 ..^cfzo 10566 ♯chash 11236 Word cword 11326 substr csubstr 11439

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-addcom 8285 ax-addass 8287 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-0id 8293 ax-rnegex 8294 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-1o 6691 df-er 6811 df-en 7027 df-dom 7028 df-fin 7029 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-inn 9314 df-n0 9575 df-z 9656 df-uz 9938 df-fz 10429 df-fzo 10567 df-ihash 11237 df-word 11327 df-substr 11440

This theorem is used by: pfxwrdsymbg 11484

W3C validator