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Theorem swrdwrdsymbg 11191

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 11178	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆))) → (𝑆 substr ⟨𝑀, 𝑁⟩) = (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))))
2	1	3expb 1228	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑀, 𝑁⟩) = (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))))
3		wrdf 11072	. . . . . . . . . 10 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
4	3	ffund 5476	. . . . . . . . 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 11074	. . . . . . . . 9 ⊢ (𝑆 ∈ Word 𝐴 → dom 𝑆 = (0..^(♯‘𝑆)))
8		elfzodifsumelfzo 10402	. . . . . . . . . . . . 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 2293	. . . . . . . . . . . 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 10217	. . . . . . . . . . 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 10217	. . . . . . . . . 10 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℤ)
23	22	ad2antrl 490	. . . . . . . . 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 10391	. . . . . . . 8 ⊢ ((𝑥 ∈ (0..^(𝑁 − 𝑀)) ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑥 + 𝑀) ∈ (𝑀..^𝑁))
26	17, 21, 24, 25	syl3anc 1271	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑥 + 𝑀) ∈ (𝑀..^𝑁))
27		funfvima 5870	. . . . . . . 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 1270	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑆‘(𝑥 + 𝑀)) ∈ (𝑆 “ (𝑀..^𝑁)))
30	29	fmpttd 5789	. . . . 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 527	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → 𝑆 ∈ Word 𝐴)
32		elfzoelz 10339	. . . . . . . . . . . . . 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 9569	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑥 + 𝑀) ∈ ℤ)
35		fvexg 5645	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑥 + 𝑀) ∈ ℤ) → (𝑆‘(𝑥 + 𝑀)) ∈ V)
36	31, 34, 35	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑆‘(𝑥 + 𝑀)) ∈ V)
37	36	ralrimiva 2603	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → ∀𝑥 ∈ (0..^(𝑁 − 𝑀))(𝑆‘(𝑥 + 𝑀)) ∈ V)
38		eqid 2229	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))) = (𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀)))
39	38	fnmpt 5449	. . . . . . . . . 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 9453	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
42		elfzel2 10215	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
43	42, 22	zsubcld 9570	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → (𝑁 − 𝑀) ∈ ℤ)
44		fzofig 10649	. . . . . . . . . . 11 ⊢ ((0 ∈ ℤ ∧ (𝑁 − 𝑀) ∈ ℤ) → (0..^(𝑁 − 𝑀)) ∈ Fin)
45	41, 43, 44	sylancr 414	. . . . . . . . . 10 ⊢ (𝑀 ∈ (0...𝑁) → (0..^(𝑁 − 𝑀)) ∈ Fin)
46	45	ad2antrl 490	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (0..^(𝑁 − 𝑀)) ∈ Fin)
47		fihashfn 11017	. . . . . . . . 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 411	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀)))) = (♯‘(0..^(𝑁 − 𝑀))))
49		fznn0sub 10249	. . . . . . . . . 10 ⊢ (𝑀 ∈ (0...𝑁) → (𝑁 − 𝑀) ∈ ℕ₀)
50		hashfzo0 11040	. . . . . . . . . 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 490	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (♯‘(0..^(𝑁 − 𝑀))) = (𝑁 − 𝑀))
53	48, 52	eqtrd 2262	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀)))) = (𝑁 − 𝑀))
54	53	oveq2d 6016	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘(𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))))) = (0..^(𝑁 − 𝑀)))
55	54	feq2d 5460	. . . . 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 490	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (𝑁 − 𝑀) ∈ ℕ₀)
58	53, 57	eqeltrd 2306	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀)))) ∈ ℕ₀)
59		iswrdinn0 11071	. . . 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 411	. . 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 2306	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑀, 𝑁⟩) ∈ Word (𝑆 “ (𝑀..^𝑁)))
62	61	3impb 1223	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆))) → (𝑆 substr ⟨𝑀, 𝑁⟩) ∈ Word (𝑆 “ (𝑀..^𝑁)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ∀wral 2508 Vcvv 2799 ⟨cop 3669 ↦ cmpt 4144 dom cdm 4718 “ cima 4721 Fun wfun 5311 Fn wfn 5312 ⟶wf 5313 ‘cfv 5317 (class class class)co 6000 Fincfn 6885 0cc0 7995 + caddc 7998 − cmin 8313 ℕ₀cn0 9365 ℤcz 9442 ...cfz 10200 ..^cfzo 10334 ♯chash 10992 Word cword 11066 substr csubstr 11172

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-addcom 8095 ax-addass 8097 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-0id 8103 ax-rnegex 8104 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4383 df-iord 4456 df-on 4458 df-ilim 4459 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-frec 6535 df-1o 6560 df-er 6678 df-en 6886 df-dom 6887 df-fin 6888 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-inn 9107 df-n0 9366 df-z 9443 df-uz 9719 df-fz 10201 df-fzo 10335 df-ihash 10993 df-word 11067 df-substr 11173

This theorem is referenced by: pfxwrdsymbg 11217

W3C validator