ofpreima2 - Mathbox for Thierry Arnoux

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Theorem ofpreima2 32609

Description: Express the preimage of a function operation as a union of preimages. This version of ofpreima 32608 iterates the union over a smaller set. (Contributed by Thierry Arnoux, 8-Mar-2018.)

**Hypotheses**
Ref	Expression
ofpreima.1	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
ofpreima.2	⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
ofpreima.3	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofpreima.4	⊢ (𝜑 → 𝑅 Fn (𝐵 × 𝐶))

**Assertion**
Ref	Expression
ofpreima2	⊢ (𝜑 → (^◡(𝐹 ∘_f 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})))

Distinct variable groups: 𝐴,𝑝 𝐷,𝑝 𝐹,𝑝 𝐺,𝑝 𝑅,𝑝 𝜑,𝑝 Allowed substitution hints: 𝐵(𝑝) 𝐶(𝑝) 𝑉(𝑝)

**Proof of Theorem ofpreima2**
Step	Hyp	Ref	Expression
1		ofpreima.1	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		ofpreima.2	. . . 4 ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
3		ofpreima.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ofpreima.4	. . . 4 ⊢ (𝜑 → 𝑅 Fn (𝐵 × 𝐶))
5	1, 2, 3, 4	ofpreima 32608	. . 3 ⊢ (𝜑 → (^◡(𝐹 ∘_f 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ (^◡𝑅 “ 𝐷)((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})))
6		inundif 4430	. . . . 5 ⊢ (((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) = (^◡𝑅 “ 𝐷)
7		iuneq1 4958	. . . . 5 ⊢ ((((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) = (^◡𝑅 “ 𝐷) → ∪ 𝑝 ∈ (((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∪ 𝑝 ∈ (^◡𝑅 “ 𝐷)((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})))
8	6, 7	ax-mp 5	. . . 4 ⊢ ∪ 𝑝 ∈ (((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∪ 𝑝 ∈ (^◡𝑅 “ 𝐷)((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)}))
9		iunxun 5043	. . . 4 ⊢ ∪ 𝑝 ∈ (((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = (∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) ∪ ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})))
10	8, 9	eqtr3i 2754	. . 3 ⊢ ∪ 𝑝 ∈ (^◡𝑅 “ 𝐷)((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = (∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) ∪ ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})))
11	5, 10	eqtrdi 2780	. 2 ⊢ (𝜑 → (^◡(𝐹 ∘_f 𝑅𝐺) “ 𝐷) = (∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) ∪ ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)}))))
12		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)))
13	12	eldifbd 3916	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ 𝑝 ∈ (ran 𝐹 × ran 𝐺))
14		cnvimass 6033	. . . . . . . . . . . . . 14 ⊢ (^◡𝑅 “ 𝐷) ⊆ dom 𝑅
15	4	fndmd 6587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑅 = (𝐵 × 𝐶))
16	14, 15	sseqtrid 3978	. . . . . . . . . . . . 13 ⊢ (𝜑 → (^◡𝑅 “ 𝐷) ⊆ (𝐵 × 𝐶))
17	16	ssdifssd 4098	. . . . . . . . . . . 12 ⊢ (𝜑 → ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐶))
18	17	sselda 3935	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐶))
19		1st2nd2 7963	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (𝐵 × 𝐶) → 𝑝 = ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩)
20		elxp6 7958	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ran 𝐹 × ran 𝐺) ↔ (𝑝 = ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ ∧ ((1^st ‘𝑝) ∈ ran 𝐹 ∧ (2^nd ‘𝑝) ∈ ran 𝐺)))
21	20	simplbi2 500	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ → (((1^st ‘𝑝) ∈ ran 𝐹 ∧ (2^nd ‘𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
22	18, 19, 21	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((1^st ‘𝑝) ∈ ran 𝐹 ∧ (2^nd ‘𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
23	13, 22	mtod 198	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ ((1^st ‘𝑝) ∈ ran 𝐹 ∧ (2^nd ‘𝑝) ∈ ran 𝐺))
24		ianor 983	. . . . . . . . 9 ⊢ (¬ ((1^st ‘𝑝) ∈ ran 𝐹 ∧ (2^nd ‘𝑝) ∈ ran 𝐺) ↔ (¬ (1^st ‘𝑝) ∈ ran 𝐹 ∨ ¬ (2^nd ‘𝑝) ∈ ran 𝐺))
25	23, 24	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (¬ (1^st ‘𝑝) ∈ ran 𝐹 ∨ ¬ (2^nd ‘𝑝) ∈ ran 𝐺))
26		disjsn 4663	. . . . . . . . 9 ⊢ ((ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅ ↔ ¬ (1^st ‘𝑝) ∈ ran 𝐹)
27		disjsn 4663	. . . . . . . . 9 ⊢ ((ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅ ↔ ¬ (2^nd ‘𝑝) ∈ ran 𝐺)
28	26, 27	orbi12i 914	. . . . . . . 8 ⊢ (((ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅) ↔ (¬ (1^st ‘𝑝) ∈ ran 𝐹 ∨ ¬ (2^nd ‘𝑝) ∈ ran 𝐺))
29	25, 28	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅))
30	1	ffnd 6653	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝐴)
31		dffn3 6664	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
32	30, 31	sylib 218	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ran 𝐹)
33	2	ffnd 6653	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn 𝐴)
34		dffn3 6664	. . . . . . . . . 10 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺:𝐴⟶ran 𝐺)
35	33, 34	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐴⟶ran 𝐺)
36	35	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝐺:𝐴⟶ran 𝐺)
37		fimacnvdisj 6702	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅) → (^◡𝐹 “ {(1^st ‘𝑝)}) = ∅)
38		ineq1 4164	. . . . . . . . . . . 12 ⊢ ((^◡𝐹 “ {(1^st ‘𝑝)}) = ∅ → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = (∅ ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})))
39		0in 4348	. . . . . . . . . . . 12 ⊢ (∅ ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅
40	38, 39	eqtrdi 2780	. . . . . . . . . . 11 ⊢ ((^◡𝐹 “ {(1^st ‘𝑝)}) = ∅ → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅)
41	37, 40	syl 17	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅) → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅)
42	41	ex 412	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶ran 𝐹 → ((ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅ → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅))
43		fimacnvdisj 6702	. . . . . . . . . . 11 ⊢ ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅) → (^◡𝐺 “ {(2^nd ‘𝑝)}) = ∅)
44		ineq2 4165	. . . . . . . . . . . 12 ⊢ ((^◡𝐺 “ {(2^nd ‘𝑝)}) = ∅ → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ ∅))
45		in0 4346	. . . . . . . . . . . 12 ⊢ ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ ∅) = ∅
46	44, 45	eqtrdi 2780	. . . . . . . . . . 11 ⊢ ((^◡𝐺 “ {(2^nd ‘𝑝)}) = ∅ → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅)
47	43, 46	syl 17	. . . . . . . . . 10 ⊢ ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅) → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅)
48	47	ex 412	. . . . . . . . 9 ⊢ (𝐺:𝐴⟶ran 𝐺 → ((ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅ → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅))
49	42, 48	jaao 956	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ 𝐺:𝐴⟶ran 𝐺) → (((ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅) → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅))
50	32, 36, 49	syl2an2r 685	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅) → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅))
51	29, 50	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅)
52	51	iuneq2dv 4966	. . . . 5 ⊢ (𝜑 → ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))∅)
53		iun0 5011	. . . . 5 ⊢ ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))∅ = ∅
54	52, 53	eqtrdi 2780	. . . 4 ⊢ (𝜑 → ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅)
55	54	uneq2d 4119	. . 3 ⊢ (𝜑 → (∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) ∪ ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)}))) = (∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) ∪ ∅))
56		un0 4345	. . 3 ⊢ (∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) ∪ ∅) = ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)}))
57	55, 56	eqtrdi 2780	. 2 ⊢ (𝜑 → (∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) ∪ ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)}))) = ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})))
58	11, 57	eqtrd 2764	1 ⊢ (𝜑 → (^◡(𝐹 ∘_f 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ∖ cdif 3900 ∪ cun 3901 ∩ cin 3902 ∅c0 4284 {csn 4577 ⟨cop 4583 ∪ ciun 4941 × cxp 5617 ^◡ccnv 5618 dom cdm 5619 ran crn 5620 “ cima 5622 Fn wfn 6477 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ∘_f cof 7611 1^st c1st 7922 2^nd c2nd 7923

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pr 5371 ax-un 7671

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-1st 7924 df-2nd 7925

This theorem is referenced by: sibfof 34308

W3C validator