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Theorem swrdwrdsymbg 11235

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 11222	. . . 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 11109	. . . . . . . . . 10 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
4	3	ffund 5483	. . . . . . . . 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 11111	. . . . . . . . 9 ⊢ (𝑆 ∈ Word 𝐴 → dom 𝑆 = (0..^(♯‘𝑆)))
8		elfzodifsumelfzo 10436	. . . . . . . . . . . . 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 10250	. . . . . . . . . . 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 10250	. . . . . . . . . 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 10425	. . . . . . . 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 5881	. . . . . . . 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 5798	. . . . 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 10372	. . . . . . . . . . . . . 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 9596	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑁 − 𝑀))) → (𝑥 + 𝑀) ∈ ℤ)
35		fvexg 5654	. . . . . . . . . . . 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 5456	. . . . . . . . . 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 9480	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
42		elfzel2 10248	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
43	42, 22	zsubcld 9597	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → (𝑁 − 𝑀) ∈ ℤ)
44		fzofig 10684	. . . . . . . . . . 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 11053	. . . . . . . . 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 10282	. . . . . . . . . 10 ⊢ (𝑀 ∈ (0...𝑁) → (𝑁 − 𝑀) ∈ ℕ₀)
50		hashfzo0 11077	. . . . . . . . . 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 6029	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘(𝑥 ∈ (0..^(𝑁 − 𝑀)) ↦ (𝑆‘(𝑥 + 𝑀))))) = (0..^(𝑁 − 𝑀)))
55	54	feq2d 5467	. . . . 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 11108	. . . 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 2800 ⟨cop 3670 ↦ cmpt 4148 dom cdm 4723 “ cima 4726 Fun wfun 5318 Fn wfn 5319 ⟶wf 5320 ‘cfv 5324 (class class class)co 6013 Fincfn 6904 0cc0 8022 + caddc 8025 − cmin 8340 ℕ₀cn0 9392 ℤcz 9469 ...cfz 10233 ..^cfzo 10367 ♯chash 11027 Word cword 11103 substr csubstr 11216

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 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-addcom 8122 ax-addass 8124 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-0id 8130 ax-rnegex 8131 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138

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 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-frec 6552 df-1o 6577 df-er 6697 df-en 6905 df-dom 6906 df-fin 6907 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209 df-le 8210 df-sub 8342 df-neg 8343 df-inn 9134 df-n0 9393 df-z 9470 df-uz 9746 df-fz 10234 df-fzo 10368 df-ihash 11028 df-word 11104 df-substr 11217

This theorem is referenced by: pfxwrdsymbg 11261

W3C validator