limciun - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > limciun		Structured version Visualization version GIF version

Theorem limciun 26014

Description: A point is a limit of 𝐹 on the finite union ∪ 𝑥 ∈ 𝐴𝐵(𝑥) iff it is the limit of the restriction of 𝐹 to each 𝐵(𝑥). (Contributed by Mario Carneiro, 30-Dec-2016.)

**Hypotheses**
Ref	Expression
limciun.1	⊢ (𝜑 → 𝐴 ∈ Fin)
limciun.2	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ ℂ)
limciun.3	⊢ (𝜑 → 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶ℂ)
limciun.4	⊢ (𝜑 → 𝐶 ∈ ℂ)

**Assertion**
Ref	Expression
limciun	⊢ (𝜑 → (𝐹 lim_ℂ 𝐶) = (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐶 𝑥,𝐹 Allowed substitution hints: 𝜑(𝑥) 𝐵(𝑥)

Proof of Theorem limciun Dummy variables 𝑔 𝑎 𝑘 𝑢 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limccl 25995	. . . 4 ⊢ (𝐹 lim_ℂ 𝐶) ⊆ ℂ
2		limcresi 26005	. . . . . 6 ⊢ (𝐹 lim_ℂ 𝐶) ⊆ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)
3	2	rgenw 3083	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 (𝐹 lim_ℂ 𝐶) ⊆ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)
4		ssiin 5016	. . . . 5 ⊢ ((𝐹 lim_ℂ 𝐶) ⊆ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) lim_ℂ 𝐶) ↔ ∀𝑥 ∈ 𝐴 (𝐹 lim_ℂ 𝐶) ⊆ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))
5	3, 4	mpbir 234	. . . 4 ⊢ (𝐹 lim_ℂ 𝐶) ⊆ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)
6	1, 5	ssini 4194	. . 3 ⊢ (𝐹 lim_ℂ 𝐶) ⊆ (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))
7	6	a1i 11	. 2 ⊢ (𝜑 → (𝐹 lim_ℂ 𝐶) ⊆ (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)))
8		elriin 5043	. . . 4 ⊢ (𝑦 ∈ (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)) ↔ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)))
9		simprl 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) → 𝑦 ∈ ℂ)
10		limciun.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ Fin)
11	10	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) → 𝐴 ∈ Fin)
12		simplrr 789	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) → ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))
13		nfcv 2927	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥𝐹
14		nfcsb1v 3879	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵
15	13, 14	nfres 5971	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥(𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵)
16		nfcv 2927	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥 lim_ℂ
17		nfcv 2927	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥𝐶
18	15, 16, 17	nfov 7430	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) lim_ℂ 𝐶)
19	18	nfcri 2919	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥 𝑦 ∈ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) lim_ℂ 𝐶)
20		csbeq1a 3869	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑥⦌𝐵)
21	20	reseq2d 5969	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝐹 ↾ 𝐵) = (𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵))
22	21	oveq1d 7415	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ((𝐹 ↾ 𝐵) lim_ℂ 𝐶) = ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) lim_ℂ 𝐶))
23	22	eleq2d 2851	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶) ↔ 𝑦 ∈ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) lim_ℂ 𝐶)))
24	19, 23	rspc 3572	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶) → 𝑦 ∈ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) lim_ℂ 𝐶)))
25	12, 24	mpan9 515	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ 𝑎 ∈ 𝐴) → 𝑦 ∈ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) lim_ℂ 𝐶))
26		limciun.3	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶ℂ)
27	26	ad2antrr 738	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶ℂ)
28		ssiun2 5008	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ 𝐴 → ⦋𝑎 / 𝑥⦌𝐵 ⊆ ∪ 𝑎 ∈ 𝐴 ⦋𝑎 / 𝑥⦌𝐵)
29		nfcv 2927	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑎𝐵
30	29, 14, 20	cbviun 4995	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑎 ∈ 𝐴 ⦋𝑎 / 𝑥⦌𝐵
31	28, 30	sseqtrrdi 3980	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ 𝐴 → ⦋𝑎 / 𝑥⦌𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
32	31	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → ⦋𝑎 / 𝑥⦌𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
33	27, 32	fssresd 6735	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → (𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵):⦋𝑎 / 𝑥⦌𝐵⟶ℂ)
34		simpr 489	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
35		limciun.2	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ ℂ)
36	35	ad2antrr 738	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝐵 ⊆ ℂ)
37		nfcv 2927	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥ℂ
38	14, 37	nfss 3932	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵 ⊆ ℂ
39	20	sseq1d 3970	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝐵 ⊆ ℂ ↔ ⦋𝑎 / 𝑥⦌𝐵 ⊆ ℂ))
40	38, 39	rspc 3572	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ⊆ ℂ → ⦋𝑎 / 𝑥⦌𝐵 ⊆ ℂ))
41	34, 36, 40	sylc 66	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → ⦋𝑎 / 𝑥⦌𝐵 ⊆ ℂ)
42		limciun.4	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ∈ ℂ)
43	42	ad2antrr 738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ ℂ)
44		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	33, 41, 43, 44	ellimc2 25997	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → (𝑦 ∈ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) lim_ℂ 𝐶) ↔ (𝑦 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝑦 ∈ 𝑢 → ∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)))))
46	45	adantlr 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ 𝑎 ∈ 𝐴) → (𝑦 ∈ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) lim_ℂ 𝐶) ↔ (𝑦 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝑦 ∈ 𝑢 → ∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)))))
47	25, 46	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ 𝑎 ∈ 𝐴) → (𝑦 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝑦 ∈ 𝑢 → ∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢))))
48	47	simprd 500	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ 𝑎 ∈ 𝐴) → ∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝑦 ∈ 𝑢 → ∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
49		simplrl 788	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ 𝑎 ∈ 𝐴) → 𝑢 ∈ (TopOpen‘ℂ_fld))
50		simplrr 789	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ 𝑎 ∈ 𝐴) → 𝑦 ∈ 𝑢)
51		rsp 3253	. . . . . . . . . . . . 13 ⊢ (∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝑦 ∈ 𝑢 → ∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)) → (𝑢 ∈ (TopOpen‘ℂ_fld) → (𝑦 ∈ 𝑢 → ∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢))))
52	48, 49, 50, 51	syl3c 67	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ 𝑎 ∈ 𝐴) → ∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢))
53	52	ralrimiva 3157	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) → ∀𝑎 ∈ 𝐴 ∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢))
54		nfv 1937	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)
55		nfcv 2927	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(TopOpen‘ℂ_fld)
56		nfv 1937	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 𝐶 ∈ 𝑘
57		nfcv 2927	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥𝑘
58		nfcv 2927	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥{𝐶}
59	14, 58	nfdif 4086	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥(⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶})
60	57, 59	nfin 4179	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))
61	15, 60	nfima 6061	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶})))
62		nfcv 2927	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝑢
63	61, 62	nfss 3932	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢
64	56, 63	nfan 1922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)
65	55, 64	nfrexw 3313	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)
66	20	difeq1d 4082	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝐵 ∖ {𝐶}) = (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))
67	66	ineq2d 4175	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑘 ∩ (𝐵 ∖ {𝐶})) = (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶})))
68	21, 67	imaeq12d 6054	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))))
69	68	sseq1d 3970	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢))
70	69	anbi2d 641	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → ((𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ (𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
71	70	rexbidv 3189	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ ∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
72	54, 65, 71	cbvralw 3307	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ ∀𝑎 ∈ 𝐴 ∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢))
73	53, 72	sylibr 237	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) → ∀𝑥 ∈ 𝐴 ∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))
74		eleq2 2854	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑔‘𝑥) → (𝐶 ∈ 𝑘 ↔ 𝐶 ∈ (𝑔‘𝑥)))
75		ineq1 4168	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑔‘𝑥) → (𝑘 ∩ (𝐵 ∖ {𝐶})) = ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶})))
76	75	imaeq2d 6053	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑔‘𝑥) → ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))))
77	76	sseq1d 3970	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑔‘𝑥) → (((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))
78	74, 77	anbi12d 643	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑔‘𝑥) → ((𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
79	78	ac6sfi 9232	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)) → ∃𝑔(𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
80	11, 73, 79	syl2anc 595	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) → ∃𝑔(𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
81	44	cnfldtop 24901	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) ∈ Top
82		frn 6703	. . . . . . . . . . . 12 ⊢ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) → ran 𝑔 ⊆ (TopOpen‘ℂ_fld))
83	82	ad2antrl 740	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ran 𝑔 ⊆ (TopOpen‘ℂ_fld))
84	11	adantr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝐴 ∈ Fin)
85		ffn 6695	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) → 𝑔 Fn 𝐴)
86	85	ad2antrl 740	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝑔 Fn 𝐴)
87		dffn4 6788	. . . . . . . . . . . . 13 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴–onto→ran 𝑔)
88	86, 87	sylib 221	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝑔:𝐴–onto→ran 𝑔)
89		fofi 9261	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑔:𝐴–onto→ran 𝑔) → ran 𝑔 ∈ Fin)
90	84, 88, 89	syl2anc 595	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ran 𝑔 ∈ Fin)
91		unicntop 24903	. . . . . . . . . . . 12 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
92	91	rintopn 23027	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ ran 𝑔 ⊆ (TopOpen‘ℂ_fld) ∧ ran 𝑔 ∈ Fin) → (ℂ ∩ ∩ ran 𝑔) ∈ (TopOpen‘ℂ_fld))
93	81, 83, 90, 92	mp3an2i 1490	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → (ℂ ∩ ∩ ran 𝑔) ∈ (TopOpen‘ℂ_fld))
94	42	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) → 𝐶 ∈ ℂ)
95	94	ad2antrr 738	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝐶 ∈ ℂ)
96		simpl 487	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → 𝐶 ∈ (𝑔‘𝑥))
97	96	ralimi 3102	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → ∀𝑥 ∈ 𝐴 𝐶 ∈ (𝑔‘𝑥))
98	97	ad2antll 741	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ∀𝑥 ∈ 𝐴 𝐶 ∈ (𝑔‘𝑥))
99		eleq2 2854	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑔‘𝑥) → (𝐶 ∈ 𝑧 ↔ 𝐶 ∈ (𝑔‘𝑥)))
100	99	ralrn 7073	. . . . . . . . . . . . 13 ⊢ (𝑔 Fn 𝐴 → (∀𝑧 ∈ ran 𝑔 𝐶 ∈ 𝑧 ↔ ∀𝑥 ∈ 𝐴 𝐶 ∈ (𝑔‘𝑥)))
101	86, 100	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → (∀𝑧 ∈ ran 𝑔 𝐶 ∈ 𝑧 ↔ ∀𝑥 ∈ 𝐴 𝐶 ∈ (𝑔‘𝑥)))
102	98, 101	mpbird 260	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ∀𝑧 ∈ ran 𝑔 𝐶 ∈ 𝑧)
103		elrint 4950	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (ℂ ∩ ∩ ran 𝑔) ↔ (𝐶 ∈ ℂ ∧ ∀𝑧 ∈ ran 𝑔 𝐶 ∈ 𝑧))
104	95, 102, 103	sylanbrc 594	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝐶 ∈ (ℂ ∩ ∩ ran 𝑔))
105		indifcom 4238	. . . . . . . . . . . . . 14 ⊢ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶})) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))
106		iunin1 5032	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))
107	105, 106	eqtr4i 2791	. . . . . . . . . . . . 13 ⊢ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶})) = ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))
108	107	imaeq2i 6051	. . . . . . . . . . . 12 ⊢ (𝐹 “ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) = (𝐹 “ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})))
109		imaiun 7233	. . . . . . . . . . . 12 ⊢ (𝐹 “ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) = ∪ 𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})))
110	108, 109	eqtri 2788	. . . . . . . . . . 11 ⊢ (𝐹 “ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) = ∪ 𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})))
111		inss2 4192	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℂ ∩ ∩ ran 𝑔) ⊆ ∩ ran 𝑔
112		fnfvelrn 7065	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑔 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ran 𝑔)
113	85, 112	sylan 591	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ran 𝑔)
114		intss1 4924	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔‘𝑥) ∈ ran 𝑔 → ∩ ran 𝑔 ⊆ (𝑔‘𝑥))
115	113, 114	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ 𝑥 ∈ 𝐴) → ∩ ran 𝑔 ⊆ (𝑔‘𝑥))
116	111, 115	sstrid 3950	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ 𝑥 ∈ 𝐴) → (ℂ ∩ ∩ ran 𝑔) ⊆ (𝑔‘𝑥))
117	116	ssdifd 4101	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ 𝑥 ∈ 𝐴) → ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔‘𝑥) ∖ {𝐶}))
118		sslin 4197	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔‘𝑥) ∖ {𝐶}) → (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})) ⊆ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶})))
119		imass2 6095	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})) ⊆ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶})) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ (𝐹 “ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶}))))
120	117, 118, 119	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ 𝑥 ∈ 𝐴) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ (𝐹 “ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶}))))
121		indifcom 4238	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶})) = (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶}))
122	121	imaeq2i 6051	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) = ((𝐹 ↾ 𝐵) “ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶})))
123		inss1 4191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶})) ⊆ 𝐵
124		resima2 6006	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶})) ⊆ 𝐵 → ((𝐹 ↾ 𝐵) “ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶}))) = (𝐹 “ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶}))))
125	123, 124	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ↾ 𝐵) “ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶}))) = (𝐹 “ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶})))
126	122, 125	eqtri 2788	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) = (𝐹 “ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶})))
127	120, 126	sseqtrrdi 3980	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ 𝑥 ∈ 𝐴) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))))
128		sstr2 3946	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) → (((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
129	127, 128	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ 𝑥 ∈ 𝐴) → (((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
130	129	adantld 495	. . . . . . . . . . . . . . 15 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ 𝑥 ∈ 𝐴) → ((𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
131	130	ralimdva 3177	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) → (∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → ∀𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
132	131	imp 411	. . . . . . . . . . . . 13 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)) → ∀𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
133	132	adantl 486	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ∀𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
134		iunss 5005	. . . . . . . . . . . 12 ⊢ (∪ 𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢 ↔ ∀𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
135	133, 134	sylibr 237	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ∪ 𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
136	110, 135	eqsstrid 3977	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → (𝐹 “ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)
137		eleq2 2854	. . . . . . . . . . . 12 ⊢ (𝑣 = (ℂ ∩ ∩ ran 𝑔) → (𝐶 ∈ 𝑣 ↔ 𝐶 ∈ (ℂ ∩ ∩ ran 𝑔)))
138		ineq1 4168	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (ℂ ∩ ∩ ran 𝑔) → (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶})) = ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶})))
139	138	imaeq2d 6053	. . . . . . . . . . . . 13 ⊢ (𝑣 = (ℂ ∩ ∩ ran 𝑔) → (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) = (𝐹 “ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))))
140	139	sseq1d 3970	. . . . . . . . . . . 12 ⊢ (𝑣 = (ℂ ∩ ∩ ran 𝑔) → ((𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ (𝐹 “ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
141	137, 140	anbi12d 643	. . . . . . . . . . 11 ⊢ (𝑣 = (ℂ ∩ ∩ ran 𝑔) → ((𝐶 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ (𝐶 ∈ (ℂ ∩ ∩ ran 𝑔) ∧ (𝐹 “ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
142	141	rspcev 3584	. . . . . . . . . 10 ⊢ (((ℂ ∩ ∩ ran 𝑔) ∈ (TopOpen‘ℂ_fld) ∧ (𝐶 ∈ (ℂ ∩ ∩ ran 𝑔) ∧ (𝐹 “ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
143	93, 104, 136, 142	syl12anc 849	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
144	80, 143	exlimddv 1958	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
145	144	expr 461	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ 𝑢 ∈ (TopOpen‘ℂ_fld)) → (𝑦 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
146	145	ralrimiva 3157	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) → ∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝑦 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
147	26	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) → 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶ℂ)
148		iunss 5005	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ℂ ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ℂ)
149	35, 148	sylibr 237	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ℂ)
150	149	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ℂ)
151	147, 150, 94, 44	ellimc2 25997	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) → (𝑦 ∈ (𝐹 lim_ℂ 𝐶) ↔ (𝑦 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂ_fld)(𝑦 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))))
152	9, 146, 151	mpbir2and 725	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) → 𝑦 ∈ (𝐹 lim_ℂ 𝐶))
153	152	ex 417	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)) → 𝑦 ∈ (𝐹 lim_ℂ 𝐶)))
154	8, 153	biimtrid 245	. . 3 ⊢ (𝜑 → (𝑦 ∈ (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)) → 𝑦 ∈ (𝐹 lim_ℂ 𝐶)))
155	154	ssrdv 3945	. 2 ⊢ (𝜑 → (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)) ⊆ (𝐹 lim_ℂ 𝐶))
156	7, 155	eqssd 3956	1 ⊢ (𝜑 → (𝐹 lim_ℂ 𝐶) = (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1563 ∃wex 1802 ∈ wcel 2145 ∀wral 3079 ∃wrex 3089 ⦋csb 3855 ∖ cdif 3904 ∩ cin 3906 ⊆ wss 3907 {csn 4585 ∩ cint 4908 ∪ ciun 4952 ∩ ciin 4953 ran crn 5653 ↾ cres 5654 “ cima 5655 Fn wfn 6520 ⟶wf 6521 –onto→wfo 6523 ‘cfv 6525 (class class class)co 7400 Fincfn 8931 ℂcc 11086 TopOpenctopn 17464 ℂ_fldccnfld 21482 Topctop 23011 lim_ℂ climc 25982

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-pm 8815 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fi 9359 df-sup 9390 df-inf 9391 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-fz 13527 df-seq 14029 df-exp 14089 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-struct 17197 df-slot 17232 df-ndx 17244 df-base 17260 df-plusg 17313 df-mulr 17314 df-starv 17315 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-rest 17465 df-topn 17466 df-topgen 17486 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-cnfld 21483 df-top 23012 df-topon 23029 df-topsp 23051 df-bases 23064 df-cnp 23346 df-xms 24438 df-ms 24439 df-limc 25986

This theorem is referenced by: limcun 26015

W3C validator