Theorem limciun 25802

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 limℂ 𝐶) = (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) limℂ 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 25783	. . . 4 ⊢ (𝐹 limℂ 𝐶) ⊆ ℂ
2		limcresi 25793	. . . . . 6 ⊢ (𝐹 limℂ 𝐶) ⊆ ((𝐹 ↾ 𝐵) limℂ 𝐶)
3	2	rgenw 3049	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 (𝐹 limℂ 𝐶) ⊆ ((𝐹 ↾ 𝐵) limℂ 𝐶)
4		ssiin 5022	. . . . 5 ⊢ ((𝐹 limℂ 𝐶) ⊆ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) limℂ 𝐶) ↔ ∀𝑥 ∈ 𝐴 (𝐹 limℂ 𝐶) ⊆ ((𝐹 ↾ 𝐵) limℂ 𝐶))
5	3, 4	mpbir 231	. . . 4 ⊢ (𝐹 limℂ 𝐶) ⊆ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) limℂ 𝐶)
6	1, 5	ssini 4206	. . 3 ⊢ (𝐹 limℂ 𝐶) ⊆ (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) limℂ 𝐶))
7	6	a1i 11	. 2 ⊢ (𝜑 → (𝐹 limℂ 𝐶) ⊆ (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) limℂ 𝐶)))
8		elriin 5048	. . . 4 ⊢ (𝑦 ∈ (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) limℂ 𝐶)) ↔ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶)))
9		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) → 𝑦 ∈ ℂ)
10		limciun.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ Fin)
11	10	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) → 𝐴 ∈ Fin)
12		simplrr 777	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) → ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))
13		nfcv 2892	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥𝐹
14		nfcsb1v 3889	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵
15	13, 14	nfres 5955	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥(𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵)
16		nfcv 2892	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥 limℂ
17		nfcv 2892	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥𝐶
18	15, 16, 17	nfov 7420	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) limℂ 𝐶)
19	18	nfcri 2884	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥 𝑦 ∈ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) limℂ 𝐶)
20		csbeq1a 3879	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑥⦌𝐵)
21	20	reseq2d 5953	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝐹 ↾ 𝐵) = (𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵))
22	21	oveq1d 7405	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ((𝐹 ↾ 𝐵) limℂ 𝐶) = ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) limℂ 𝐶))
23	22	eleq2d 2815	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶) ↔ 𝑦 ∈ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) limℂ 𝐶)))
24	19, 23	rspc 3579	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶) → 𝑦 ∈ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) limℂ 𝐶)))
25	12, 24	mpan9 506	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ 𝑎 ∈ 𝐴) → 𝑦 ∈ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) limℂ 𝐶))
26		limciun.3	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶ℂ)
27	26	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶ℂ)
28		ssiun2 5014	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ 𝐴 → ⦋𝑎 / 𝑥⦌𝐵 ⊆ ∪ 𝑎 ∈ 𝐴 ⦋𝑎 / 𝑥⦌𝐵)
29		nfcv 2892	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑎𝐵
30	29, 14, 20	cbviun 5003	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑎 ∈ 𝐴 ⦋𝑎 / 𝑥⦌𝐵
31	28, 30	sseqtrrdi 3991	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ 𝐴 → ⦋𝑎 / 𝑥⦌𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
32	31	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → ⦋𝑎 / 𝑥⦌𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
33	27, 32	fssresd 6730	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → (𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵):⦋𝑎 / 𝑥⦌𝐵⟶ℂ)
34		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
35		limciun.2	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ ℂ)
36	35	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝐵 ⊆ ℂ)
37		nfcv 2892	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥ℂ
38	14, 37	nfss 3942	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵 ⊆ ℂ
39	20	sseq1d 3981	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝐵 ⊆ ℂ ↔ ⦋𝑎 / 𝑥⦌𝐵 ⊆ ℂ))
40	38, 39	rspc 3579	. . . . . . . . . . . . . . . . . 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) limℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → ⦋𝑎 / 𝑥⦌𝐵 ⊆ ℂ)
42		limciun.4	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐶 ∈ ℂ)
43	42	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ ℂ)
44		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
45	33, 41, 43, 44	ellimc2 25785	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ 𝑎 ∈ 𝐴) → (𝑦 ∈ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) limℂ 𝐶) ↔ (𝑦 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦 ∈ 𝑢 → ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)))))
46	45	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ 𝑎 ∈ 𝐴) → (𝑦 ∈ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) limℂ 𝐶) ↔ (𝑦 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦 ∈ 𝑢 → ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)))))
47	25, 46	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ 𝑎 ∈ 𝐴) → (𝑦 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦 ∈ 𝑢 → ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢))))
48	47	simprd 495	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ 𝑎 ∈ 𝐴) → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦 ∈ 𝑢 → ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
49		simplrl 776	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ 𝑎 ∈ 𝐴) → 𝑢 ∈ (TopOpen‘ℂfld))
50		simplrr 777	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ 𝑎 ∈ 𝐴) → 𝑦 ∈ 𝑢)
51		rsp 3226	. . . . . . . . . . . . 13 ⊢ (∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦 ∈ 𝑢 → ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)) → (𝑢 ∈ (TopOpen‘ℂfld) → (𝑦 ∈ 𝑢 → ∃𝑘 ∈ (TopOpen‘ℂfld)(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) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ 𝑎 ∈ 𝐴) → ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢))
53	52	ralrimiva 3126	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) → ∀𝑎 ∈ 𝐴 ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢))
54		nfv 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑎∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)
55		nfcv 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(TopOpen‘ℂfld)
56		nfv 1914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 𝐶 ∈ 𝑘
57		nfcv 2892	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥𝑘
58		nfcv 2892	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥{𝐶}
59	14, 58	nfdif 4095	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥(⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶})
60	57, 59	nfin 4190	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))
61	15, 60	nfima 6042	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶})))
62		nfcv 2892	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝑢
63	61, 62	nfss 3942	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢
64	56, 63	nfan 1899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)
65	55, 64	nfrexw 3289	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)
66	20	difeq1d 4091	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝐵 ∖ {𝐶}) = (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))
67	66	ineq2d 4186	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (𝑘 ∩ (𝐵 ∖ {𝐶})) = (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶})))
68	21, 67	imaeq12d 6035	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))))
69	68	sseq1d 3981	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → (((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢))
70	69	anbi2d 630	. . . . . . . . . . . . 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 3158	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
72	54, 65, 71	cbvralw 3282	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ ∀𝑎 ∈ 𝐴 ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ ⦋𝑎 / 𝑥⦌𝐵) “ (𝑘 ∩ (⦋𝑎 / 𝑥⦌𝐵 ∖ {𝐶}))) ⊆ 𝑢))
73	53, 72	sylibr 234	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) → ∀𝑥 ∈ 𝐴 ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))
74		eleq2 2818	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑔‘𝑥) → (𝐶 ∈ 𝑘 ↔ 𝐶 ∈ (𝑔‘𝑥)))
75		ineq1 4179	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑔‘𝑥) → (𝑘 ∩ (𝐵 ∖ {𝐶})) = ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶})))
76	75	imaeq2d 6034	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑔‘𝑥) → ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))))
77	76	sseq1d 3981	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑔‘𝑥) → (((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))
78	74, 77	anbi12d 632	. . . . . . . . . . 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 9238	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑘 ∧ ((𝐹 ↾ 𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)) → ∃𝑔(𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀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) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) → ∃𝑔(𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
81	44	cnfldtop 24678	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂfld) ∈ Top
82		frn 6698	. . . . . . . . . . . 12 ⊢ (𝑔:𝐴⟶(TopOpen‘ℂfld) → ran 𝑔 ⊆ (TopOpen‘ℂfld))
83	82	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ran 𝑔 ⊆ (TopOpen‘ℂfld))
84	11	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝐴 ∈ Fin)
85		ffn 6691	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝐴⟶(TopOpen‘ℂfld) → 𝑔 Fn 𝐴)
86	85	ad2antrl 728	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝑔 Fn 𝐴)
87		dffn4 6781	. . . . . . . . . . . . 13 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴–onto→ran 𝑔)
88	86, 87	sylib 218	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝑔:𝐴–onto→ran 𝑔)
89		fofi 9269	. . . . . . . . . . . 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) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ran 𝑔 ∈ Fin)
91		unicntop 24680	. . . . . . . . . . . 12 ⊢ ℂ = ∪ (TopOpen‘ℂfld)
92	91	rintopn 22803	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ ran 𝑔 ⊆ (TopOpen‘ℂfld) ∧ ran 𝑔 ∈ Fin) → (ℂ ∩ ∩ ran 𝑔) ∈ (TopOpen‘ℂfld))
93	81, 83, 90, 92	mp3an2i 1468	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → (ℂ ∩ ∩ ran 𝑔) ∈ (TopOpen‘ℂfld))
94	42	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) → 𝐶 ∈ ℂ)
95	94	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝐶 ∈ ℂ)
96		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → 𝐶 ∈ (𝑔‘𝑥))
97	96	ralimi 3067	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → ∀𝑥 ∈ 𝐴 𝐶 ∈ (𝑔‘𝑥))
98	97	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ∀𝑥 ∈ 𝐴 𝐶 ∈ (𝑔‘𝑥))
99		eleq2 2818	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑔‘𝑥) → (𝐶 ∈ 𝑧 ↔ 𝐶 ∈ (𝑔‘𝑥)))
100	99	ralrn 7063	. . . . . . . . . . . . 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) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀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 257	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ∀𝑧 ∈ ran 𝑔 𝐶 ∈ 𝑧)
103		elrint 4956	. . . . . . . . . . 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) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝐶 ∈ (ℂ ∩ ∩ ran 𝑔))
105		indifcom 4249	. . . . . . . . . . . . . 14 ⊢ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶})) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))
106		iunin1 5039	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))
107	105, 106	eqtr4i 2756	. . . . . . . . . . . . 13 ⊢ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶})) = ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))
108	107	imaeq2i 6032	. . . . . . . . . . . 12 ⊢ (𝐹 “ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) = (𝐹 “ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})))
109		imaiun 7222	. . . . . . . . . . . 12 ⊢ (𝐹 “ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) = ∪ 𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})))
110	108, 109	eqtri 2753	. . . . . . . . . . 11 ⊢ (𝐹 “ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) = ∪ 𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})))
111		inss2 4204	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℂ ∩ ∩ ran 𝑔) ⊆ ∩ ran 𝑔
112		fnfvelrn 7055	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑔 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ran 𝑔)
113	85, 112	sylan 580	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ran 𝑔)
114		intss1 4930	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔‘𝑥) ∈ ran 𝑔 → ∩ ran 𝑔 ⊆ (𝑔‘𝑥))
115	113, 114	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥 ∈ 𝐴) → ∩ ran 𝑔 ⊆ (𝑔‘𝑥))
116	111, 115	sstrid 3961	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥 ∈ 𝐴) → (ℂ ∩ ∩ ran 𝑔) ⊆ (𝑔‘𝑥))
117	116	ssdifd 4111	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥 ∈ 𝐴) → ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔‘𝑥) ∖ {𝐶}))
118		sslin 4209	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔‘𝑥) ∖ {𝐶}) → (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶})) ⊆ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶})))
119		imass2 6076	. . . . . . . . . . . . . . . . . . 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‘ℂfld) ∧ 𝑥 ∈ 𝐴) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ (𝐹 “ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶}))))
121		indifcom 4249	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶})) = (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶}))
122	121	imaeq2i 6032	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) = ((𝐹 ↾ 𝐵) “ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶})))
123		inss1 4203	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶})) ⊆ 𝐵
124		resima2 5990	. . . . . . . . . . . . . . . . . . . 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 2753	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) = (𝐹 “ (𝐵 ∩ ((𝑔‘𝑥) ∖ {𝐶})))
127	120, 126	sseqtrrdi 3991	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥 ∈ 𝐴) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))))
128		sstr2 3956	. . . . . . . . . . . . . . . . 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‘ℂfld) ∧ 𝑥 ∈ 𝐴) → (((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
130	129	adantld 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥 ∈ 𝐴) → ((𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
131	130	ralimdva 3146	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝐴⟶(TopOpen‘ℂfld) → (∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → ∀𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
132	131	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)) → ∀𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
133	132	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ∀𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
134		iunss 5012	. . . . . . . . . . . 12 ⊢ (∪ 𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢 ↔ ∀𝑥 ∈ 𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
135	133, 134	sylibr 234	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀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 3988	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → (𝐹 “ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)
137		eleq2 2818	. . . . . . . . . . . 12 ⊢ (𝑣 = (ℂ ∩ ∩ ran 𝑔) → (𝐶 ∈ 𝑣 ↔ 𝐶 ∈ (ℂ ∩ ∩ ran 𝑔)))
138		ineq1 4179	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (ℂ ∩ ∩ ran 𝑔) → (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶})) = ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶})))
139	138	imaeq2d 6034	. . . . . . . . . . . . 13 ⊢ (𝑣 = (ℂ ∩ ∩ ran 𝑔) → (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) = (𝐹 “ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))))
140	139	sseq1d 3981	. . . . . . . . . . . 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 632	. . . . . . . . . . 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 3591	. . . . . . . . . 10 ⊢ (((ℂ ∩ ∩ ran 𝑔) ∈ (TopOpen‘ℂfld) ∧ (𝐶 ∈ (ℂ ∩ ∩ ran 𝑔) ∧ (𝐹 “ ((ℂ ∩ ∩ ran 𝑔) ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
143	93, 104, 136, 142	syl12anc 836	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ∈ (𝑔‘𝑥) ∧ ((𝐹 ↾ 𝐵) “ ((𝑔‘𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
144	80, 143	exlimddv 1935	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦 ∈ 𝑢)) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
145	144	expr 456	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) ∧ 𝑢 ∈ (TopOpen‘ℂfld)) → (𝑦 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
146	145	ralrimiva 3126	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
147	26	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) → 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶ℂ)
148		iunss 5012	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ℂ ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ℂ)
149	35, 148	sylibr 234	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ℂ)
150	149	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ℂ)
151	147, 150, 94, 44	ellimc2 25785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) → (𝑦 ∈ (𝐹 limℂ 𝐶) ↔ (𝑦 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦 ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶 ∈ 𝑣 ∧ (𝐹 “ (𝑣 ∩ (∪ 𝑥 ∈ 𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))))
152	9, 146, 151	mpbir2and 713	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶))) → 𝑦 ∈ (𝐹 limℂ 𝐶))
153	152	ex 412	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ ℂ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ ((𝐹 ↾ 𝐵) limℂ 𝐶)) → 𝑦 ∈ (𝐹 limℂ 𝐶)))
154	8, 153	biimtrid 242	. . 3 ⊢ (𝜑 → (𝑦 ∈ (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) limℂ 𝐶)) → 𝑦 ∈ (𝐹 limℂ 𝐶)))
155	154	ssrdv 3955	. 2 ⊢ (𝜑 → (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) limℂ 𝐶)) ⊆ (𝐹 limℂ 𝐶))
156	7, 155	eqssd 3967	1 ⊢ (𝜑 → (𝐹 limℂ 𝐶) = (ℂ ∩ ∩ 𝑥 ∈ 𝐴 ((𝐹 ↾ 𝐵) limℂ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  ⦋csb 3865   ∖ cdif 3914   ∩ cin 3916   ⊆ wss 3917  {csn 4592  ∩ cint 4913  ∪ ciun 4958  ∩ ciin 4959  ran crn 5642   ↾ cres 5643   “ cima 5644   Fn wfn 6509  ⟶wf 6510  –onto→wfo 6512  ‘cfv 6514  (class class class)co 7390  Fincfn 8921  ℂcc 11073  TopOpenctopn 17391  ℂ_fldccnfld 21271  Topctop 22787   lim_ℂ climc 25770

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fi 9369  df-sup 9400  df-inf 9401  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-q 12915  df-rp 12959  df-xneg 13079  df-xadd 13080  df-xmul 13081  df-fz 13476  df-seq 13974  df-exp 14034  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-struct 17124  df-slot 17159  df-ndx 17171  df-base 17187  df-plusg 17240  df-mulr 17241  df-starv 17242  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-rest 17392  df-topn 17393  df-topgen 17413  df-psmet 21263  df-xmet 21264  df-met 21265  df-bl 21266  df-mopn 21267  df-cnfld 21272  df-top 22788  df-topon 22805  df-topsp 22827  df-bases 22840  df-cnp 23122  df-xms 24215  df-ms 24216  df-limc 25774

This theorem is referenced by:  limcun  25803