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Theorem limciun 25056

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 25037	. . . 4 ⊢ (𝐹 lim_ℂ 𝐶) ⊆ ℂ
2		limcresi 25047	. . . . . 6 ⊢ (𝐹 lim_ℂ 𝐶) ⊆ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)
3	2	rgenw 3078	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 (𝐹 lim_ℂ 𝐶) ⊆ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)
4		ssiin 4990	. . . . 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 230	. . . 4 ⊢ (𝐹 lim_ℂ 𝐶) ⊆ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)
6	1, 5	ssini 4171	. . 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 5015	. . . 4 ⊢ (𝑦 ∈ (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)) ↔ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)))
9		simprl 768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) → 𝑦 ∈ ℂ)
10		limciun.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ Fin)
11	10	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) → 𝐴 ∈ Fin)
12		simplrr 775	. . . . . . . . . . . . . . . 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 2909	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥𝐹
14		nfcsb1v 3862	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵
15	13, 14	nfres 5892	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥(𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵)
16		nfcv 2909	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥 lim_ℂ
17		nfcv 2909	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥𝐶
18	15, 16, 17	nfov 7301	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) lim_ℂ 𝐶)
19	18	nfcri 2896	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥 𝑦 ∈ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) lim_ℂ 𝐶)
20		csbeq1a 3851	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑥⦌𝐵)
21	20	reseq2d 5890	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝐹 ↾ 𝐵) = (𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵))
22	21	oveq1d 7286	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ((𝐹 ↾ 𝐵) lim_ℂ 𝐶) = ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) lim_ℂ 𝐶))
23	22	eleq2d 2826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶) ↔ 𝑦 ∈ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) lim_ℂ 𝐶)))
24	19, 23	rspc 3548	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶) → 𝑦 ∈ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) lim_ℂ 𝐶)))
25	12, 24	mpan9 507	. . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶ℂ)
28		ssiun2 4982	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ 𝐴 → ⦋𝑎 / 𝑥⦌𝐵 ⊆ ∪ 𝑎 ∈ 𝐴 ⦋𝑎 / 𝑥⦌𝐵)
29		nfcv 2909	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑎𝐵
30	29, 14, 20	cbviun 4971	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑎 ∈ 𝐴 ⦋𝑎 / 𝑥⦌𝐵
31	28, 30	sseqtrrdi 3977	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ 𝐴 → ⦋𝑎 / 𝑥⦌𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
32	31	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → ⦋𝑎 / 𝑥⦌𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
33	27, 32	fssresd 6639	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → (𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵):⦋𝑎 / 𝑥⦌𝐵⟶ℂ)
34		simpr 485	. . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝐵 ⊆ ℂ)
37		nfcv 2909	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥ℂ
38	14, 37	nfss 3918	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵 ⊆ ℂ
39	20	sseq1d 3957	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝐵 ⊆ ℂ ↔ ⦋𝑎 / 𝑥⦌𝐵 ⊆ ℂ))
40	38, 39	rspc 3548	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ⊆ ℂ → ⦋𝑎 / 𝑥⦌𝐵 ⊆ ℂ))
41	34, 36, 40	sylc 65	. . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ ℂ)
44		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	33, 41, 43, 44	ellimc2 25039	. . . . . . . . . . . . . . . 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 712	. . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . 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 496	. . . . . . . . . . . . 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 774	. . . . . . . . . . . . 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 775	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂ_fld) ∧ 𝑦 ∈ 𝑢)) ∧ 𝑎 ∈ 𝐴) → 𝑦 ∈ 𝑢)
51		rsp 3132	. . . . . . . . . . . . 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 66	. . . . . . . . . . . 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 3110	. . . . . . . . . . 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 1921	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)
55		nfcv 2909	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(TopOpen‘ℂ_fld)
56		nfv 1921	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 𝐶 ∈ 𝑘
57		nfcv 2909	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥𝑘
58		nfcv 2909	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥{𝐶}
59	14, 58	nfdif 4065	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥(⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶})
60	57, 59	nfin 4156	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))
61	15, 60	nfima 5976	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶})))
62		nfcv 2909	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝑢
63	61, 62	nfss 3918	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢
64	56, 63	nfan 1906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)
65	55, 64	nfrex 3240	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∃𝑘 ∈ (TopOpen‘ℂ_fld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)
66	20	difeq1d 4061	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝐵 ∖ {𝐶}) = (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))
67	66	ineq2d 4152	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑘 ∩ (𝐵 ∖ {𝐶})) = (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶})))
68	21, 67	imaeq12d 5969	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))))
69	68	sseq1d 3957	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢))
70	69	anbi2d 629	. . . . . . . . . . . . 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 3228	. . . . . . . . . . . 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 3372	. . . . . . . . . . 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 233	. . . . . . . . . 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 2829	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑔‘𝑥) → (𝐶 ∈ 𝑘 ↔ 𝐶 ∈ (𝑔‘𝑥)))
75		ineq1 4145	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑔‘𝑥) → (𝑘 ∩ (𝐵 ∖ {𝐶})) = ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶})))
76	75	imaeq2d 5968	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑔‘𝑥) → ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))))
77	76	sseq1d 3957	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑔‘𝑥) → (((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))
78	74, 77	anbi12d 631	. . . . . . . . . . 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 9036	. . . . . . . . . 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 584	. . . . . . . . 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 23945	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) ∈ Top
82		frn 6605	. . . . . . . . . . . 12 ⊢ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) → ran 𝑔 ⊆ (TopOpen‘ℂ_fld))
83	82	ad2antrl 725	. . . . . . . . . . 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 481	. . . . . . . . . . . 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 6598	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝐴⟶(TopOpen‘ℂ_fld) → 𝑔 Fn 𝐴)
86	85	ad2antrl 725	. . . . . . . . . . . . 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 6692	. . . . . . . . . . . . 13 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴–onto→ran 𝑔)
88	86, 87	sylib 217	. . . . . . . . . . . 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 9083	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑔:𝐴–onto→ran 𝑔) → ran 𝑔 ∈ Fin)
90	84, 88, 89	syl2anc 584	. . . . . . . . . . 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 23947	. . . . . . . . . . . 12 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
92	91	rintopn 22056	. . . . . . . . . . 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 1465	. . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) → 𝐶 ∈ ℂ)
95	94	ad2antrr 723	. . . . . . . . . . 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 483	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → 𝐶 ∈ (𝑔‘𝑥))
97	96	ralimi 3089	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → ∀𝑥 ∈ 𝐴 𝐶 ∈ (𝑔‘𝑥))
98	97	ad2antll 726	. . . . . . . . . . . 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 2829	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑔‘𝑥) → (𝐶 ∈ 𝑧 ↔ 𝐶 ∈ (𝑔‘𝑥)))
100	99	ralrn 6961	. . . . . . . . . . . . 13 ⊢ (𝑔 Fn 𝐴 → (∀𝑧 ∈ ran 𝑔 𝐶 ∈ 𝑧 ↔ ∀𝑥 ∈ 𝐴 𝐶 ∈ (𝑔‘𝑥)))
101	86, 100	syl 17	. . . . . . . . . . . 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 256	. . . . . . . . . . 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 4928	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (ℂ ∩ ∩ ran 𝑔) ↔ (𝐶 ∈ ℂ ∧ ∀𝑧 ∈ ran 𝑔 𝐶 ∈ 𝑧))
104	95, 102, 103	sylanbrc 583	. . . . . . . . . 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 4212	. . . . . . . . . . . . . 14 ⊢ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶})) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))
106		iunin1 5006	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))
107	105, 106	eqtr4i 2771	. . . . . . . . . . . . 13 ⊢ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶})) = ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))
108	107	imaeq2i 5966	. . . . . . . . . . . 12 ⊢ (𝐹 “ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) = (𝐹 “ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})))
109		imaiun 7115	. . . . . . . . . . . 12 ⊢ (𝐹 “ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) = ∪ 𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})))
110	108, 109	eqtri 2768	. . . . . . . . . . 11 ⊢ (𝐹 “ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) = ∪ 𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})))
111		inss2 4169	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℂ ∩ ∩ ran 𝑔) ⊆ ∩ ran 𝑔
112		fnfvelrn 6955	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑔 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ran 𝑔)
113	85, 112	sylan 580	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ran 𝑔)
114		intss1 4900	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔‘𝑥) ∈ ran 𝑔 → ∩ ran 𝑔 ⊆ (𝑔‘𝑥))
115	113, 114	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ 𝑥 ∈ 𝐴) → ∩ ran 𝑔 ⊆ (𝑔‘𝑥))
116	111, 115	sstrid 3937	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ 𝑥 ∈ 𝐴) → (ℂ ∩ ∩ ran 𝑔) ⊆ (𝑔‘𝑥))
117	116	ssdifd 4080	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ 𝑥 ∈ 𝐴) → ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔‘𝑥) ∖ {𝐶}))
118		sslin 4174	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔‘𝑥) ∖ {𝐶}) → (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})) ⊆ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶})))
119		imass2 6009	. . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ 𝑥 ∈ 𝐴) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ (𝐹 “ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶}))))
121		indifcom 4212	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶})) = (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶}))
122	121	imaeq2i 5966	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) = ((𝐹 ↾ 𝐵) “ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶})))
123		inss1 4168	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶})) ⊆ 𝐵
124		resima2 5925	. . . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) = (𝐹 “ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶})))
127	120, 126	sseqtrrdi 3977	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂ_fld) ∧ 𝑥 ∈ 𝐴) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))))
128		sstr2 3933	. . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . . . 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 3105	. . . . . . . . . . . . . 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 407	. . . . . . . . . . . . 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 482	. . . . . . . . . . . 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 4980	. . . . . . . . . . . 12 ⊢ (∪ 𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢 ↔ ∀𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
135	133, 134	sylibr 233	. . . . . . . . . . 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 3974	. . . . . . . . . 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 2829	. . . . . . . . . . . 12 ⊢ (𝑣 = (ℂ ∩ ∩ ran 𝑔) → (𝐶 ∈ 𝑣 ↔ 𝐶 ∈ (ℂ ∩ ∩ ran 𝑔)))
138		ineq1 4145	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (ℂ ∩ ∩ ran 𝑔) → (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶})) = ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶})))
139	138	imaeq2d 5968	. . . . . . . . . . . . 13 ⊢ (𝑣 = (ℂ ∩ ∩ ran 𝑔) → (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) = (𝐹 “ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))))
140	139	sseq1d 3957	. . . . . . . . . . . 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 631	. . . . . . . . . . 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 3561	. . . . . . . . . 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 834	. . . . . . . . 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 1942	. . . . . . . 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 457	. . . . . . 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 3110	. . . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) → 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶ℂ)
148		iunss 4980	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ℂ ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ℂ)
149	35, 148	sylibr 233	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ℂ)
150	149	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ℂ)
151	147, 150, 94, 44	ellimc2 25039	. . . . . 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 710	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶))) → 𝑦 ∈ (𝐹 lim_ℂ 𝐶))
153	152	ex 413	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)) → 𝑦 ∈ (𝐹 lim_ℂ 𝐶)))
154	8, 153	syl5bi 241	. . 3 ⊢ (𝜑 → (𝑦 ∈ (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)) → 𝑦 ∈ (𝐹 lim_ℂ 𝐶)))
155	154	ssrdv 3932	. 2 ⊢ (𝜑 → (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) lim_ℂ 𝐶)) ⊆ (𝐹 lim_ℂ 𝐶))
156	7, 155	eqssd 3943	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 205 ∧ wa 396 = wceq 1542 ∃wex 1786 ∈ wcel 2110 ∀wral 3066 ∃wrex 3067 ⦋csb 3837 ∖ cdif 3889 ∩ cin 3891 ⊆ wss 3892 {csn 4567 ∩ cint 4885 ∪ ciun 4930 ∩ ciin 4931 ran crn 5591 ↾ cres 5592 “ cima 5593 Fn wfn 6427 ⟶wf 6428 –onto→wfo 6430 ‘cfv 6432 (class class class)co 7271 Fincfn 8716 ℂcc 10870 TopOpenctopn 17130 ℂ_fldccnfld 20595 Topctop 22040 lim_ℂ climc 25024

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-pm 8601 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fi 9148 df-sup 9179 df-inf 9180 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-fz 13239 df-seq 13720 df-exp 13781 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-struct 16846 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-mulr 16974 df-starv 16975 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-rest 17131 df-topn 17132 df-topgen 17152 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cnp 22377 df-xms 23471 df-ms 23472 df-limc 25028

This theorem is referenced by: limcun 25057

W3C validator